a. Equilibrium ice area

If the loss of MY ice area over the course of a year—equal to or, equivalently, —is exactly balanced by the area of FY ice surviving the melt season—equal to f—then the total area at the summer minimum remains unchanged and the system is in equilibrium. At equilibrium, (1)–(3) become

and

The ratio in (7) is of particular importance because s and W can be readily estimated from passive microwave satellite data. A further useful relation is the ratio of MY to FY ice at the summer minimum as it depends only on the survivability of MY ice from one summer to the next :

We can evaluate how well these relations hold with the results of the CICE hindcast. Using 1979–2006 as an example of a mean state, we approximate the equilibrium value of each variable by its average value over that period (see Table 1). The left-hand side of (7), s/W = 0.374, is in good agreement with the right-hand side, . The left-hand side of (8), m/f = 1.60, agrees well with the right-hand side, .

The previous paragraph shows that (7) and (8) hold to within a few percent if we approximate the equilibrium of each variable by its mean over 28 yr. We will show in the following section that the area is never long out of equilibrium and the equilibrium area is well estimated by an average over a few years. In addition, we assume that the ice survival ratios are a reflection of Arctic climate, which may contain trends.

The sensitivity of s/W to changes in the ice survival ratios is examined by plotting its dependence on α and β as given by (7). This sensitivity, given by the slope of the surface in Fig. 5, depends critically on the location of the mean state in α − β space (see appendix A for full representations of the slopes). In particular, for a given value of s/W, if β is large and α is small (case A in Figs. 1 and 5), then s/W is very sensitive to changes in the survival ratios. However, if β and α are comparable (case B in Figs. 1 and 5), then s/W is relatively insensitive to changes in the survival ratios. This poses a challenge for simulating changes to the summer minimum ice area in warming scenarios: there are many combinations of , and that can provide a realistic simulation of the climatological seasonal cycle of ice area, and each combination has a unique sensitivity to a changing climate. The results of the CICE hindcast are shown in Fig. 2 and by point C in Fig. 5.

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Plot of , which is (7) under the approximation (from Table 1, the survival ratios of MY ice through the growth season and melt season are similar), over the regime . Points A–C denote different regions on the surface with the same value of s/W but different values of α and β. Point C is the result of the CICE hindcast 28-yr mean (α = 0.203 and β = 0.602 from Table 1). Point A is α ≈ 0.1 and β ≈ 0.82. Point B is α ≈ 0.3 and β ≈ 0.36.

Citation: Journal of Climate 24, 9; 10.1175/2010JCLI3823.1

The sensitivities of f and m to changes in the survival ratios can be evaluated by use of (5) and (6) (see appendix A). These sensitivities reflect that if α decreases, then f decreases because less FY ice survives the summer melt season and m decreases because less FY ice is available to replenish the MY ice category each year. If or decreases, then m decreases because less MY ice survives the growth or melt seasons, respectively, but there is an increase in f because more of the Arctic becomes available for FY ice to grow over the winter (see Figs. 3c and 4). In the CICE hindcast, the FY and MY ice survival ratios decrease at comparable rates, resulting in a large reduction of the MY ice area while the September FY ice area remains relatively constant (Table 1 and Fig. 3b). These trends in FY and MY ice areas are consistent with recent observations (e.g., Kwok et al. 2009).

The results of the CICE hindcast indicate that the Arctic sea ice system is in a regime where the total summer minimum ice area s is more sensitive to changes in than in (see appendix A and consider the values in Table 1). This arises because FY ice may grow in place of the MY ice that is lost during the growth season. Also, s is more sensitive to changes in α than in either or (see point C in Fig. 5). This will become increasingly so as the ratio of MY ice area to FY ice area continues to decrease in a warming climate.

b. Equilibrium ice volume

Trends in ice area can be related to trends in ice volume through the reduced model. In equilibrium, the average sea ice thickness at the summer minimum is

and the ice volume is

Equation (9) shows that the average ice thickness is independent of α. This can equivalently be expressed as , meaning that the fractional decline in volume is equal to the fractional decline in area under decreasing α. However, for a decrease in either or , there will be a larger fractional decline in volume than in area provided that (see appendix A). This arises from the larger area loss of the thick MY ice than the loss of total ice area under a decrease in either of the MY ice survival ratios. Additionally, υ can decrease due to the thinning of ice within the FY and MY categories (Fig. 4), while s is unchanged with respect to changes in and (at constant FY and MY ice survivability). Therefore, a larger percent decrease in volume than area will occur under global warming scenarios. This can be seen in the CICE hindcast (from Table 1, υ declined at an average rate of −21% decade⁻¹ while s declined at −11% decade⁻¹, with respect to their average values over 1979–2006) as well as in simulations with GCMs (over the 70 yr of CO₂ ramping in the CCSM3 simulation, shown in Fig. 6, υ declined at an average rate of −16% decade⁻¹ while s declined at −12% decade⁻¹, with respect to their average values over that period).

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Arctic September sea ice area and volume from a CCSM3 simulation with 1% yr⁻¹ CO₂ ramping between the years 0 and 70. Two runs are shown over the years 0–70. To illustrate the trends under CO₂ ramping, the 2 runs have been averaged together over this period, and a 20-yr running mean has been applied to this average (thick black line).

Citation: Journal of Climate 24, 9; 10.1175/2010JCLI3823.1

c. Interannual variability in ice area

We now consider how the mean survivabilities of FY and MY ice—which set the total summer ice area and partitioning between ice types—control the response of the system to interannual climate variability. The deviation of summer minimum ice area from its equilibrium value in year n is

where we have dropped terms that are second order in the perturbed quantities.² The persistence of area perturbations from one summer (s′_n−1) to the next summer (s′_n) is regulated by the quantity multiplying s′_n−1 in (11), which is large when and are large and α is small. Thus, persistence is high when FY ice survivability and MY ice survivability are different from one another (case A in Figs. 1 and 5) and low when they are comparable (case B in Figs. 1 and 5).

To the extent that the perturbations in the survival ratios and the winter maximum area are described by white noise, (11) is equivalent to a first-order autoregressive [AR(1)] process [see appendix B and von Storch and Zwiers (1999) for the properties of these systems]. Then, the characteristic response time scale, defined as the e-folding time over which the ice retains information about a perturbation in the summer minimum area, is given by

For the CICE hindcast, this time scale as calculated from the average survival ratios (Table 1) is 1.2 yr. Because τ_s is relatively small, the area is never out of equilibrium for long and averaging over only a few years is a sufficient approximation to the current equilibrium state.

This time scale can also be estimated directly from the time series of the minimum ice area from the CICE hindcast. If we assume that the minimum area is an AR(1) process, then τ_s ≈ −(1 year/lnR), where R = 0.24 is the autocorrelation at lag 1 yr of the linearly detrended time series. This gives τ_s ≈ 0.7 yr, which differs slightly from our first estimate because in (12) we ignored the small correlations among the perturbations.³ Both estimates of the time scale indicate low year-to-year memory in the minimum sea ice area, consistent with observations (Blanchard-Wrigglesworth et al. 2011). We note that although our analysis is limited to the period 1979–2006, our estimate of memory is qualitatively consistent with the observed September ice extent in more recent years: following the record minimum extent in 2007, subsequent years (2008–10) show a slight recovery and are consistent with variability about the long-term linear trend.

Which variables most influence the year-to-year perturbations in summer minimum ice area can be determined by considering the correlation of each variable in (11) with s_n′. After linearly detrending the time series of each variable in the CICE hindcast, we find that W′_n is correlated at R = 0.1, at R = 0.02, and s′_n−1 at R = 0.24. Because α′_n and are significantly correlated with each other (R = 0.63), we cannot determine which is most responsible for the variations in s′_n. However, together they account for nearly all of the variance in the summer minimum ice area, with α′_n correlated with s′_n at R = 0.82 and at R = 0.75. The survival ratios α′_n and are driven by stochastic weather conditions during the melt season, so it is likely that area anomalies from before about April do not provide predictability for the summer minimum area.

Modeling studies find increasing variability in the summer minimum ice area in a warming climate (Holland et al. 2009; Goosse et al. 2009). From (11), if the average ice survival ratios , and continue to decline, α′_n will become increasingly dominant in its control of s′_n. From the CICE hindcast, the variances in α′ and are comparable. It is therefore likely that increasing variability in the survivability of FY ice, as opposed to simply the transition to an Arctic that is dominated by FY ice, is the source of increasing variability in the summer minimum ice area as simulated by GCMs under greenhouse warming.

d. Interannual variability in ice volume

The perturbations in the summer minimum ice volume about its equilibrium value are given by

where the terms not shown are with respect to the perturbations . If these perturbations are well approximated by white-noise processes, the memory time scale for volume is then

which is longer than that for area (τ_s) provided that . This longer time scale arises because the perturbations in the summer minimum ice volume are dominantly controlled by the perturbations in the thicker MY ice, which carries with it the memory of the sea ice anomalies in the previous year. The persistence time scale for volume is likely longer than τ_υ because in (14) we have ignored the additional memory of the previous year’s volume that arises through the persistence of MY ice thickness anomalies, [see Bitz et al. (1996) and L’Heveder and Houssais (2001) for autoregressive models of sea ice thickness].

This time scale, estimated from the average values of the variables from the CICE hindcast (Table 1) is ≈2.7 yr, which is longer than the corresponding estimate for τ_s. The longer memory time scale for volume than area is also apparent in the CCSM3 simulation (over years −150 to 0 in Fig. 6, τ_s ≈ 1.4 yr and τ_υ ≈ 7.0 yr). Corresponding to this longer memory time scale is the tendency for ice volume to remain out of equilibrium for much longer periods of time than does area. In contrast to ice area, it is difficult to accurately estimate the equilibrium ice volume with an average over even a large number of years in a volume time series, and care must be taken when determining whether a given change in volume over a short period of time necessarily implies a change in the equilibrium state of the system.

a. The dependence of Arctic sea ice on FY and MY ice survivability

Through the use of a reduced model for Arctic sea ice and a simulation of sea ice conditions over the period 1979–2006, we have explored the ways in which the survival ratios of FY and MY ice control the summer minimum ice area and volume. The results of the CICE hindcast suggest that perturbations in summer minimum FY and MY ice areas are comparable in magnitude and, so, contribute about equally to perturbations in the total minimum area (Fig. 3b). The perturbation analysis showed that variability in summer ice area is dominated by perturbations in the survival ratios α and β^m, which are governed mostly by stochastic weather noise through the melt season. Therefore, prediction of the summer minimum ice area in a given year depends more on the accurate prediction of weather conditions through the melt season than on the accurate representation of the FY and MY ice area anomalies that exist before the melt season.

Under decreasing ice survival ratios, a larger decrease in summer MY ice area than FY ice area will occur (see Fig. 3b and appendix A). As a result, trends in the total summer ice area are due more to changes in the amount of MY ice surviving the year than to changes in the amount of FY ice surviving the summer. This is consistent with the trend under decreasing MY ice survival ratios toward an Arctic that is increasingly dominated by FY ice [see (8)]. The CICE hindcast also indicates that Arctic sea ice is in a regime where the ice area is particularly sensitive to changes in the FY ice survival ratio, α (Fig. 5). It is therefore critical that trends in FY ice and its survivability are observed and modeled accurately.

b. Memory, variability, and mean state sensitivity

We have found that September ice area and volume behave approximately as AR(1) processes. There are several properties of AR(1) systems that are relevant to understanding the current state of the Arctic sea ice system and what changes can be expected in a warming climate. First, systems with long memory time scales recover more slowly from perturbations than do systems with short memory time scales and, therefore, exhibit greater variability in response to a forcing of particular variance (see appendix B). Second, systems with longer response times have enhanced sensitivity to long-term forcing compared to systems with shorter response times (Hansen et al. 1985; Bitz and Roe 2004; also see appendix A).

The relatively short time scale τ_s for Arctic sea ice area, arising from the survival of ice that grows over the winter, corresponds to a mean state that has relatively little variability about its long-term trend and low sensitivity to trends in the ice survival ratios. The persistence time scale for sea ice volume (τ_υ) is longer than that for ice area, contributing to a relatively greater variance in ice volume than area (see Fig. 6). This longer memory time scale for volume corresponds to a greater volume sensitivity than area sensitivity to trends in the survival ratios (see appendix A). Relatively larger reductions in ice volume than area have been shown to occur in the CICE hindcast (Table 1), in GCMs (Fig. 6; Gregory et al. 2002), and in observations (Kwok et al. 2009).

Because the September ice area—averaged over only a few years—is at all times very nearly in equilibrium, the observed decrease in Arctic sea ice area is a clear indicator of a changing climate. Conversely, the relatively long time scale τ_υ implies that the September sea ice volume can be out of equilibrium with climate forcing for long periods of time. Thus, while sea ice volume has been shown to be relatively more sensitive to changes in climate forcing than is ice area, its use as an indicator of changing climate conditions is complicated by its long memory time scale; any statistical test for significance of a trend in ice volume is limited by a reduced number of degrees of freedom (Flato 1994).

c. How does changing memory affect the trajectory of sea ice decline?

Given the strong thickness–growth feedback of sea ice (Bitz and Roe 2004)—where in a warming climate we can expect the thicker MY ice to thin at a greater rate than the thinner FY ice—and the fact that the ratio of MY to FY ice entering into the MY ice category each year is decreasing [see (8)], it is likely that the difference between FY and MY ice survival ratios will decrease in a warming climate. If this occurs, the Arctic sea ice system would move toward a regime of decreased memory and decreased sensitivity to climate forcing (consider Fig. 5, or the equations in appendix A, as and ). Indeed, a decrease in the quantity is seen to occur over the CICE hindcast, implying a reduction in memory and mean state sensitivity over the course of the simulation (see Table 1). The loss of MY ice in the observed Arctic sea ice system (Johannessen et al. 1999; Comiso 2002; Nghiem et al. 2007; Maslanik et al. 2007; Kwok et al. 2009) suggests that the system may be undergoing a similar decrease in memory.

If the memory time scale and mean state sensitivity of Arctic sea ice decrease sufficiently quickly under a warming climate, a slowing in the rate of area and volume loss could occur. This is consistent with the characteristic trajectory of September sea ice area decline in twenty-first century simulations where the rate of change of Arctic sea ice area decreases late in the simulation despite a continued increase in climate forcing (e.g., Fig. 6). An inflection point in ice decline could also occur if during the warming the ice survival ratios decrease quickly for some time and then slow in their rate of decrease at some later time. It is the interaction between the trends in the survival ratios (the forcing) and the memory time scale (the sensitivity to forcing) that determines the trajectory of the ice decline, and further study with an FY ice tracer within fully coupled climate models is needed to determine the exact reason for the inflection point seen in simulations. To establish whether such a trajectory is likely in the observed sea ice system, estimates of the mean state and trends in Arctic sea ice survival ratios from observations should be compared with those of coupled simulations that exhibit such behavior.

The interpretation of sea ice area and volume as AR(1) processes is useful for gaining insight into the current state and recent trends in the Arctic sea ice system. Its use for long-term projections is dependent upon the assumption that the perturbations in FY and MY survivability that force the system continue to be well approximated by white noise.

d. New metrics for improving sea ice projections

Stroeve et al. (2007) compare observed Arctic sea ice trends to the results of the models participating in the Intergovernmental Panel on Climate Change’s Fourth Assessment Report (IPCC AR4) under the Special Report on Emissions Scenarios’ (SRES) A1B emissions scenario. Of the models with a mean Arctic ice extent within 20% of the observed ice extent over the period 1953–95, the multimodel mean trend in September ice extent over the period 1979–2006 was −4.3 ± 0.3% decade⁻¹—considerably less than the trend in observations of −9.1 ± 1.5% decade⁻¹ over that period. This suggests that while an accurate reproduction of the seasonal cycle of ice extent is necessary, it is not sufficient to reproduce the correct sensitivity to changing climate conditions.

The findings of Stroeve et al. (2007) are consistent with our result that it is possible to accurately model the climatological seasonal cycle of sea ice area and volume without correctly representing the mean state in FY–MY ice area and thickness space (given by ) and its corresponding sensitivity to climate forcing (see Figs. 1 and 5). For example, a sea ice model that tends to homogenize FY and MY ice physics by not resolving t^f and t^m effectively lowers the MY ice survival ratio while raising the FY ice survival ratio. Such a model, though it may reproduce the seasonal cycle of the ice area, will nonetheless have decreased memory time scales (particularly for volume) and be generally less sensitive to climate forcing. Indeed, two models that are among the most sophisticated [the National Center for Atmospheric Research’s (NCAR) CCSM3 and Met Office’s (UKMO) Hadley Centre Global Environmental Model (HadGEM)] most closely reproduce the observed trend in September Arctic sea ice extent (Stroeve et al. 2007).

If we are to make accurate projections of September sea ice area and volume trends, it is important that sea ice models are validated using more than just the observed area and volume. Given the connection between memory time scale and mean state sensitivity, one metric for models might be to correctly reproduce the variance and lag 1 yr autocorrelations from observations of summer minimum ice area and volume. However, given the relatively short time series of observed ice area and volume, there are large uncertainties associated with using these values to estimate the underlying mean state of the sea ice system. A more useful metric for establishing skill in sea ice projections would be the direct comparison of FY and MY ice survival ratios between models and observations. While observations of Arctic sea ice are now available for such a comparison (e.g., Maslanik et al. 2007; Kwok et al. 2009), the models participating in the IPCC AR4 have not traced FY and MY ice individually. Thus, we have a valuable opportunity to validate models with observations that have not previously been considered in the modeling process.