## 1. Introduction

Recent submarine-based measurements indicate that the thickness of sea ice in some parts of the central Arctic has decreased at a remarkably high rate over the past few decades (Rothrock et al. 1999; Wadhams and Davis 2000). Based on their analysis of sonar reflections from the ice draft, Rothrock et al. (1999) concluded that the ice thinned by about 1.3 m in the 1990s relative to the period 1958–76 by comparing all overlapping submarine tracks between the two periods. A further intriguing aspect of these data is that the measured thinning is a strong function of the draft. Figure 1 shows the reduction in ice draft for each location where a submarine track from the 1990s crossed a track from the period 1958–76, as a function of the mean ice draft in the same location in 1958–76, based on data from Rothrock et al. (1999). The reduction in draft increases roughly linearly with the initial draft.

This relationship is in striking contrast to the situation on seasonal time scales, where the largest change in the mass balance of sea ice occurs for thin ice. Thin ice tends to experience high melt rates in summer due to ice–albedo feedback, and thin ice may also grow rapidly in winter when it can conduct a great deal of heat from the ocean to the atmosphere. Clearly the thickness dependence of the thinning on decadal time scales shown in Fig. 1 is not due to ice–albedo feedback, nor does Fig. 1 show that the reduction in thickness is merely limited by the ice available to melt.

Several recent studies argue that changes in the atmospheric circulation are chiefly responsible for the thickness change measured by the submarines (Tucker et al. 2001; Holloway and Sou 2002; Rothrock et al. 2003). Holloway and Sou (2002) hypothesized that the thickness change averaged over the limited area measured by submarines is larger than the basin average owing to a redistribution of ice by surface winds. Their model results show ice diverged out of the region sampled by the submarines and converged near the Canadian Archipalego, outside of the submarine data boundary. Some modelers find that the Arctic sea ice volume is more responsive to recent wind stress anomalies than to recent atmospheric heat flux anomalies (Zhang et al. 2000; Holloway and Sou 2002). If recent basin-scale (∼1000 km) thickness change is primarily due to winds, it is natural to ask if the subbasin-scale distribution of thickness change (i.e., on the scale of the separation of the 29 submarine crossings, or ∼100 km and smaller) is also due to winds. If so, is the apparent thickness dependence in Fig. 1 due to changes in the winds as well? Unfortunately sea ice models are only moderately successful at reproducing the spatial pattern of the thickness in a given year (Rothrock et al. 2003, see their Fig. 7) or the spatial pattern of thickness change measured by submarines (Holloway and Sou 2002), so this question cannot be easily answered.

In this study we consider an alternative explanation: we show that the inherent properties of sea ice give rise to a strongly thickness dependent sensitivity to perturbations, whether due to anomalous wind or ocean stresses or surface heat fluxes. We show that coupled climate models exhibit roughly the same relationship between ice thinning and the initial thickness, as shown in Fig. 1, when subject to increasing levels of carbon dioxide in the models' atmosphere. These models have a wide variety of physical formulations for sea ice (some are motionless while others have ice dynamics that account for the material properties of ice), but they all represent the same basic thermodynamics of sea ice. With this motivation, we present a mechanism based only on the thermodynamics of sea ice to explain how sea ice can thin most where the ice was initially thickest. We also develop a general theory, which accounts for ice dynamics as well as thermodynamics, to describe the equilibrium thickness sensitivity of sea ice to changes in forcing.

## 2. Ice thickness change in CMIP models

We first present the ice thickness change simulated by three coupled climate models, each with very different sea ice physics from the others. Each model was forced identically with increasing CO_{2} at the rate of 1% yr^{−1} as part of the Coupled Model Intercomparison Project (CMIP; Meehl et al. 2000). We found similar results in each of the 14 CMIP models that we examined, but selected just 3 to demonstrate our point. The first model is the Community Climate System Model version 2 (CCSM2), which has the most sophisticated sea ice physics: Ice dynamics are computed assuming a viscous plastic rheology, a subgrid-scale ice thickness distribution is parameterized with explicit deformation and redistribution, and heat storage in brine pockets is computed explicitly (Briegleb et al. 2002). The second model is the Hadley Centre Climate Model version 3 (HadCM3). The sea ice in HadCM3 (Gordon et al. 2000) has an intermediate level of complexity among the three models, with ice motion derived solely from ocean currents and thermodynamics based on the Semtner (1976) three-layer method. The Max-Planck-Institut fuer Meteorologie Model version 3 (ECHAM3-LSG) has the simplest sea ice, which is motionless and neglects heat storage in ice altogether (Cubasch et al. 1997).

A more complete presentation of the simulations of Arctic climate change in the CMIP models is given by Holland and Bitz (2003) and Flato (2004). Here we analyze the models specifically to see if they reproduce the relationship between thinning and initial thickness that is apparent from submarine measurements.

Flato (2004, manuscript submitted to *Climate Dyn.*) noted that the spatial pattern of thinning in global climate models tends to have a maximum somewhere in the central Arctic. This is illustrated for each of the three models in Fig. 2 (left column), which shows that the zonal mean thickness in the control (present day CO_{2} level) and perturbed (at twice the present day CO_{2} level) simulations. In all cases the greatest thinning occurs where the ice is thickest, which in the zonal mean lies in the central Arctic.

We also show scatterplots (right column in Fig. 2) of the spatial distribution of the reduction in thickness as a function of the control thickness for the three models. The points in Fig. 2 are taken for all grid cells north of 70°N to roughly represent the central Arctic, but the scatterplots for all grid cells in the Northern Hemisphere (not shown) look similar. The scatterplots resemble Fig. 1 in that the thickness reduction in the models also increases with initial thickness (i.e., from the control case). The global average of the radiative forcing increase from doubling CO_{2}, which was prescribed in the perturbed case of the CMIP models, is about three and a half times greater than estimates of the direct radiative forcing increase that has occurred in the interval between the submarine measurements (based on the work of Hansen et al. 2002), and yet the thickness change for a given thickness is about the same in the models and as measured by the submarines. This suggests that the models appear to miss some aspect of the recent change. One thing we know for certain is that the trends in the surface winds are much weaker in most CMIP models compared to recent observed trends (Gillett et al. 2002). In spite of the likely differences in the source of the thickness perturbation (i.e., surface radiative fluxes in the CMIP models and mostly wind stress in nature), the relationship between the resulting thickness change and the thickness is nearly the same.

The thickness change and control thickness shown in Fig. 2 are taken from annual mean output from the models, while the submarine data in Fig. 1 are from September. Because the same basic relationship emerges from each, we infer that, with regard to the physics of the thickness change, there is probably nothing special about September in the observations. Furthermore the relationship between the change and initial state extends to first year ice (ice that does not survive summer melt) in the models.

## 3. Sea ice growth as a feedback on thickness

The common behavior among the CMIP models, some of which have motionless sea ice, suggests that the basic thermodynamics of sea ice alone must play a dominant role. The mechanism we propose follows a simple line of reasoning. It has long been known that, when perturbed, sea ice returns to its equilibrium (or quasi-equilibrium) thickness by adjusting its growth rate (see e.g., Untersteiner 1961, 1964). The rate at which sea ice can respond to a perturbation in forcing (i.e., the response time) depends on the growth adjustment process. Using a simplified analytical model, Thorndike (1992) found that the response time of sea ice is about 3 yr for 1-m-thick ice and he noted that the response time increases with ice thickness. Here we take these ideas one step further to consider the thickness dependence of the equilibrium thickness change in response to a perturbation in the climate forcing. These results are directly analogous to Hansen et al. (1985), who showed that systems with long response times also experience a large equilibrium temperature change in response to radiative forcing. If sea ice behaves in a similar way, both the equilibrium thickness change and the response time should increase with thickness.

We begin by considering the mass budget of a sea ice floe. The ice attains an equilibrium thickness *h*_{eq} when the annually accumulated growth *G* and melt *M* balance under climatological mean conditions, provided that the floe remains within a similar climatic region long enough. Theoretical efforts to compute the equilibrium thickness can lead to rather complex equations (see e.g., Kolesnikov 1958). Instead, we use the simplified analytic model of Thorndike (1992), which includes only the barest elements of the climate system in order to allow tractable analytic expressions for *G* and *M.* This model neglects the motion of sea ice altogether, thereby assuming that the ice can always establish an equilibrium thickness. In the following sections we consider the errors due to some of the assumptions made for this analytical model by analyzing the behavior of sea ice in more complex numerical models.

*G*and

*M*by assumming that they are proportional to the sum of the net flux incident on the top surface and the ocean heat flux incident at the base of the ice. He approximated the net surface flux from the seasonal mean and fixed the length of the growth and melt seasons each at one-half year. A schematic of the system is given in Fig. 3, from which Thorndike's equations follow:

*T*= 0°C is assumed in Eq. (2). A list of variable definitions and default values is given in Table 1. Note that with these assumptions, only

*G*is a function of

*h*through its dependence on the surface temperature in winter:

*G*and

*M*taken from Eqs. (1–3) for two hypothetical “latitudes,” which are distinguished soley by varying the mean flux of surface insolation

*F*

_{SW}(175 W m

^{−2}for 80°N and 200 W m

^{−2}for 68°N). We show curves for both a control climate, using the default values given in Table 1, and a perturbed climate, where

*A*is decreased such that Δ

*A*= −11.3 W m

^{−2}. The latter is meant to crudely approximate doubling CO

_{2}by increasing the downwelling longwave radiation on the annual mean at the surface by

*G*and

*M*defines

*h*

_{eq}for the control climate. It also shows the change in

*G*and

*M*due to the increase in radiative forcing in the perturbed case and the subsequent reduction in equilibrium thickness by Δ

*h*

_{eq}that is required to establish a new balance between growth and melt.

Comparing the curves for the two hypothetical latitudes, Δ*h*_{eq} is seen to be larger for the latitude with larger equilibrium thickness. This result is due to the thickness dependence of the slope of *G* at *h*_{eq}. Where the slope is steep, the ice need not thin very much to increase the annual growth and reestablish an equilibrium. In contrast, where the slope is shallow, the ice must thin comparably more to increase the annual growth. In other words, Δ*h*_{eq} depends on the reciprocal of ∂*G*/∂*h* evaluated at the equilibrium thickness for the control climate *h _{o}*. Thorndike (1992) recognized that the response time for sea ice to adjust to equilibrium after a sudden change in thickness is equal to the reciprocal of ∂

*G*/∂

*h*evaluated at

*h*

_{eq}. Hence Δ

*h*

_{eq}and the response time depend on

*h*in a similar way.

The curves in Fig. 4 approximately agree with estimates for *G* and *M* for ice with *h*_{eq} = 3 m given by Untersteiner (1961) and Maykut (1986) based on field data and modeling.^{1} However, following Thorndike (1992), we have assumed that *M* is independent, of *h,* while Untersteiner (1961) and Maykut (1986) argue that *M* should increase if the ice becomes thin, for example, below 1 m, to account for the thickness dependence of the ice albedo. Fixing the length of the growth and melt seasons is another assumption that deserves further scrutiny. In reality the season length depends on both ice thickness and the climate conditions. Owing to the simplifying assumptions made for this analysis, the quantitative predictions should not be taken too seriously, especially below 1 m.

*h*

_{eq}as a function of

*h*. This calculation yields a theoretical curve for comparison with the reduction in ice draft from the submarine data shown in Fig. 1 and the thickness change in the CMIP models (none of which are strictly in equilibrium, but can be considered to be in quasi-equilibrium because the response time for sea ice thickness change is short compared to the time scale of change of the forcing). When subject to a change in radiative forcing, sea ice reaches a new equilibrium by adjusting

_{o}*h*such that Δ

*G*= Δ

*M.*An expansion in the dependent variables

*h*and

*A*that is linear in the change in thickness and the radiative perturbation gives

*h*

_{eq}is taken as positive for a reduction in ice thickness. Thus

*h*

_{eq}approximately increases with the reciprocal of ∂

*G*/∂

*h*(i.e., the response time). Indeed the dashed curve in Fig. 5 shows the thickness dependence of Δ

*h*

_{eq}from (∂

*G*/∂

*h*)

^{−1}alone by letting

*h*= 3 m for the term within the brackets. With this approximation, Δ

_{o}*h*

_{eq}is quadratic in

*h*.

_{o}In Fig. 4, we distinguished latitudes by only varying *F*_{SW} and hence neglected the spatial dependence of *D.* Continuing with this approximation, we can apply Eq. (7) to relate the spatial distribution of thinning to the initial thickness, as we did for the observed submarine data and CMIP model output. The magnitude of our estimate in Fig. 5 is not meant to be directly comparable to the CMIP models in Fig. 2 because the prescribed longwave forcing perturbation does not include feedbacks from other parts of the system, such as from clouds or water vapor.

*h*

_{eq}

*λ*

_{h}

*F*

*λ*

_{h}is a

*thickness*sensitivity parameter. Equation (8) is analogous to the relation Δ

*T*

_{eq}=

*λ*Δ

*F,*where

*λ*is known as the

*climate*sensitivity parameter (see, e.g., Houghton et al. 2001, p. 216) and is a measure of the ratio of the temperature change to the perturbative radiative forcing. By analogy,

*λ*

_{h}is a measure of the of the ratio of the thickness change to the perturbative forcing.

Feedback processes alter the magnitude of *λ* and *λ*_{h}. Others have recognized that the stabilizing effect of growth on ice thickness is a feedback process (e.g., Gordon and O'Farrell 1997; Zhang et al. 2000; L'Heveder and Houssais 2001). Indeed, the growth– thickness relationship yields a feedback that is as basic to sea ice thickness as radiative emission is to temperature. However, the temperature dependence of the climate sensitivity parameter due to radiative emission alone, *λ*_{0} = (4*ϵσT*^{3})^{−1} (*ϵ* is earth's emissivity and *σ* is the Stefan–Boltzmann constant), is usually ignored because the temperature dependence of *λ*_{0} is not substantial for the range of temperatures usually found on earth. In contrast, according to Fig. 5, the thickness dependence of *λ*_{h} from the growth–thickness relationship alone varies by at least a factor of 2 for thicknesses in the range of 1–6 m, typical of ice observed in the Arctic.

*λ*

_{h}. Nonetheless, the thickness sensitivity parameter can easily be generalized to account for the thickness dependence of

*M,*as well as broader possibilities for the influence of radiative forcing on

*G*and

*M*(such as changes to the length of the melt or growth season):

*M*increase for small

*h,*compared to the expression for Δ

*h*

_{eq}in Eq. (7) (also see Fig. 5), Δ

*h*

_{eq}would increase for small

*h.*In the next section, we will present results from a numerical model that explicitly allows for ice–albedo feedback and other ice thermodynamics that were neglected here. In the section following that we discuss the implication of allowing ice dynamics as well.

## 4. Growth feedback in a simple numerical model

Here we recompute *G* and *M* using a numerical model that takes into account some of the important processes that were neglected by the analytic model. In particular, we include a prognostic equation for the seasonal cycle of the energy balance of sea ice, a thickness-dependent albedo parameterization, a mixed layer ocean, and variable heat transport in the atmosphere and ocean. For simplicity while exploring the thermodynamic mechanism outlined in the previous section, we use a model that lacks sea ice dynamics.

In keeping with the simple treatment of sea ice, we selected a model that lacks internal (and hence interannual) variability, the seasonally varying Energy Balance Model (EBM) of North and Coakley (1979), to which we added a prognostic equation for sea ice thickness, which is described in the appendix. This zonal-mean model has fractions of land and ocean at each latitude and is forced with a seasonal distribution of solar heat entering the top of the atmosphere. Our parameterization for the planetary albedo over sea ice depends on *h* for *h* < 1 m.

For our control case, we use model equations and parameters from North and Coakley (1979) except as noted in the appendix. We also ran the model with longwave radiation increased by 4 W m^{−2} to mimic doubling CO_{2}, which we refer to as the perturbed case. We use 200 grid points evenly spaced in sine of latitude with mixed implicit and explicit numerical schemes to allow a time step of 5 days.

Figure 6 illustrates the strong thickness dependence of the response time in the EBM when initialized with zero-thickness sea ice. In agreement with Thorndike (1992), thicker ice takes longer to reach its equilibrium thickness. These figures also illustrate that the amount the ice grows in a single year is a strong function of thickness. In contrast, the amount the ice melts in a single year is not very sensitive to thickness, although it does depend on latitude via the shortwave radiation.

Figure 7 shows estimates of *G* and *M* for 80° and 68°N derived from the EBM (see figure caption for an explanation). These curves resemble their counterparts from the analytic method, although the ice is somewhat thinner in the EBM. Under the perturbed conditions the ice at 68°N becomes seasonal, so the intersection of *G* and *M* curves occurs at their endpoints. Even with a thickness-dependent albedo parameterization, there is only a small increase in *M* for ice thinner than 1 m at 68°N. There is a small cusp near the endpoint of *M* at 68°N under the perturbed forcing as the ice becomes seasonal and there is less ice available to melt. Otherwise, for a given latitude and longwave forcing, *M* depends only weakly on thickness.

In the EBM, the change in *M* at a given latitude due to increased radiative forcing [i.e., ∂*M*/∂*F* in Eq. (9)] is quite different for the two latitudes, and yet this possibility was neglected by the analytic model. We find that ∂*M*/∂*F* is much larger at 68° than 80°N in the EBM, as the albedo decreases and the ice becomes seasonal. In addition, storage of heat in the mixed layer and warming of nearby land at 68°N prolong the melt season by several weeks. In contrast, ∂*M*/∂*F* is very small at 80°N due to the reduction in the atmospheric and oceanic heat flux into the polar cap in a warmer climate. Thus ∂*M*/∂*F* in Eq. (9) is a function of latitude (and hence *h*) in the EBM. This dependence somewhat reduces the sensitivity of Δ*h*_{eq} to *h _{o}* compared to our estimates from the analytic model.

Figure 8 shows the thickness in the control and perturbed simulation as a function of latitude. In agreement with the CMIP models (see Fig. 2), the largest thickness reduction occurs within the central Arctic. Figure 8 also shows the thickness reduction as a function of the initial thickness from the EBM along with the theoretical estimate from Eq. (7). Here Δ*h*_{eq} increases with *h _{o}* but to a lesser extent than the theoretical curve, as expected due to ice–albedo feedback and other neglected physics in the analytic model.

The results from the EBM suggest that neglected processes in the analytic model can have a sizeable influence, although the growth–thickness feedback still dominates the results. In the discussion that follows we consider the influence of ice dynamics.

## 5. Discussion of the role of ice dynamics

*O*from a given region (i.e., the volume divergence averaged over the region) in our volume budget (see Hibler and Hutchings 2002). Hence

*G*−

*M*−

*O*= 0 defines a quasi-equilibrium state for sea ice in motion. Our equation for the thickness sensitivity parameter then becomes

*λ*

_{h}within the first brackets (including the exponent) is the response time, which depends on the thickness dependence of the volume budget terms. The component within the second brackets represents the response of the volume budget terms to the climate forcing.

In an investigation of the potential for multiple equilibria in sea ice, Hibler and Hutchings (2002) estimated the thickness dependence of *O* averaged over the whole Arctic basin using an idealized dynamic–thermodynamic sea ice model. They found that *O* is roughly linear in *h* for *h* < 2 m, when the ice can be considered rather weak,^{2} and *O* reaches a maximum at about 3 m and decreases for larger *h,* as the ice strength begins to noticeably build with *h.* If we assume that this estimate for *O* also applies on, say, the spatial area of the submarine crossing or the grid area in the CMIP models, then the contribution from ∂*O*/∂*h,* roughly speaking, would reduce Δ*h*_{eq} for *h* < 2 m but increase it for *h* > 4 m. Therefore, the thickness dependence of *O* would make Δ*h*_{eq} increase even more steeply with *h _{o}*, compared to our estimates from either the analytic model or the EBM.

Recent trends in the ice circulation (see, e.g., Steele et al. 1996; Tucker et al. 2001; Rigor et al. 2002) due to trends in the atmospheric (or ocean) circulation can be considered part of the response to climate forcing and hence can be accounted for in our equations with ∂*O*/∂*F.* One can imagine that a trend in the winds or ocean stresses could be correlated in space with the pattern of sea ice thickness, which could, either by coincidence or due to some kind of coupled response, yield a thickness dependence in ∂*O*/∂*F.* While we know of no study that has shown estimates of ∂*O*/∂*F* as a function of *h,* sea ice hindcasts with models forced by observed atmospheric conditions indicate that the recent thickness change can be mainly attributed to trends in the winds (Zhang et al. 2000; Holloway and Sou 2002). According to Eq. (10), ∂*O*/∂*F* multiplies the portion of Δ*h*_{eq} that is due to the thickness dependence of *G* and the other terms in the volume budget, and the thickness dependence of this latter portion was the main focus of this study.

## 6. Conclusions

We have shown that the spatial distribution of sea ice thinning over the past few decades in the central Arctic as measured by submarines was a strong function of the initial draft. A similar relationship is reproduced individually by CMIP models when sea ice thins in response to increased levels of CO_{2} in the models' atmosphere. Moreover, because many of the CMIP models have only weak trends in their surface winds and several models lack ice dynamics altogether, the explanation of the models' behavior likely rests on ice thermodynamics.

We described a mechanism to explain the thickness dependence of the thickness change in response to a radiative perturbation by building upon an existing theory of sea ice mass balance from Thorndike (1992). One of the predictions from Thorndike is that the rate at which sea ice can respond to a perturbation of its thickness will depend upon the growth adjustment process. The growth–thickness relationship defines a feedback process that rapidly returns thin ice to its equilibrium thickness when it is subject to a single abrupt thickness perturbation, but leads to thicker ice spending more time away from its equilibrium. When subject to an increase in radiative forcing, the annual melt will increase and annual growth will decrease. In response, the ice must thin until the annual growth balances the annual melt again. Because the growth rate of thin ice increases very rapidly when the ice thins further, thin ice need not thin much to attain a new equilibrium thickness, while thicker ice must thin more. Using simple expressions for annual growth and melt from Thorndike (1992), we estimated a theoretical relationship for the thickness change, which we found to increase approximately quadratically with thickness.

One might imagine that a strong positive feedback, such as ice–albedo feedback, would dominate the thickness dependence of the response to radiative perturbations. But ice–albedo feedback ought to cause thin ice to melt more, which is the opposite thickness dependence to what is observed by submarines and simulated in CMIP models. Compared to our estimates for the growth–thickness feedback alone, adding ice–albedo feedback tends to slightly level out the slope of the curve relating the thickness change to the initial thickness.

We compared our theoretical estimates to results from a simple zonal-mean energy balance model that explicitly included sea ice with ice–albedo feedback but lacked ice dynamics. The growth–thickness feedback controls the overall behavior, although ice–albedo feedback and variable season lengths have a nontrivial influence on the model results.

We generalized our theoretical framework to account for both sea ice dynamics and climate perturbations that include trends in the wind or ocean stress. Based on estimates from Hibler and Hutchings (2002), we argued that ice dynamics would likely reduce the thickness change for ice that is in free drift, or nearly so, and increase the thickness change for ice that has considerable compressive strength. This transition can be associated with an ice thickness of about 3 m. Thus, generally, ice dynamics would steepen the slope of the curve relating the thickness change to the initial thickness, compared to the relationship predicted with simple growth–thickness feedback alone.

In agreement with several other studies, comparing the rate of thinning observed by the submarines with the thinning in response to doubling CO_{2} in CMIP models suggests that recent trends in the atmospheric circulations were critical in explaining the submarines measurements. In spite of the likely differences in the source of the thickness perturbation (i.e., surface radiative fluxes in the CMIP models and mostly wind stress in the submarine measurements), the thinning increases sharply with thickness in both the CMIP models and in submarine measurements. The results from the CMIP model suggest that sea ice thinning in the future will be a strong function of thickness in response to increased levels of carbon dioxide in the atmosphere, even if trends in the atmospheric circulation do not continue.

Our results indicate that, when a climate perturbation causes the ice to thin, the greatest thinning occurs in the central Arctic, where the ice is thickest, and less thinning occurs in the subpolar seas. An estimate of the overall thickness change in the Arctic should take into account the strong thickness dependence of the change. For this reason, it is important to accurately measure and simulate the ice thickness to better understand the past and future of the Arctic ice pack.

## Acknowledgments

We thank Drs. Rothrock and Flato and one anonymous reviewer for their constructive reviews. We thank Drs. Gregory and Moritz for helpful comments and suggestions. We thank Dr. G. Schmidt for providing direct radiative forcing estimates for the twentieth century from Hansen et al. (2002). We acknowledge participating CMIP modeling groups for providing data on sea ice thickness and extent to the CMIP model archive. CMB gratefully acknowledges the support of the National Science Foundation, Office of Polar Programs Grant OPP0084287.

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## APPENDIX

### Energy Balance Model with Sea Ice

*x*is the sine of latitude,

*k*is the conductivity,

*Q*is the solar heating,

*T*

_{s}is the surface temperature,

*T*

_{b}= −2°C is the freezing point of sea water,

*α*

_{s}is the planetary albedo over sea ice,

*A*+

*BT*

_{s}is the net outgoing infrared radiation,

*D*is the diffusivity [taken to be a function of

*x*following Lindzen and Farrell (1977)],

*f*

_{W}is the fraction of the latitude covered by ocean,

*ν*controls land–ocean heat exchange, and

*T*

_{L}is the temperature of the fraction of the latitude covered by land. Note that parameters

*A,*

*B,*and

*D*used for the EBM are similar but not equivalent to the one used for the analytic calculations in section 3. The surface temperature

*T*

_{s}is constrained to be at or below

*T*

_{b}. The last term of the right-hand side of Eq. (A1) represents the land–ocean coupling in the EBM, which is necessary in a seasonal model (North and Coakley 1979). The coefficient

*η*is used to separate the meridional heat flux into atmospheric and oceanic portions in the presence of sea ice only, where we let the top surface of the ice receive 95% and the other 5% is applied to the base of the ice, as the variable

*F*

_{W}in Eq. (A2). These percentiles are based on estimates at 60°N and 60°S from the work of Trenberth and Caron (2001).

*F*

_{net}is equal to the right-hand side of Eq. (A1). We make a special calculation any time the ocean temperature drops below the

*T*

_{b}point, where we grow just enough ice to bring the temperature back to

*T*

_{b}, and adjust

*h*accordingly. By assuming the vertical temperature in the ice is always linear, the heat capacity of sea ice is neglected in this model.

*α*

_{L,W}

*x*

^{2}

*α*

_{L}= 0.6) when

*T*

_{L}≤ −2°C or south of 65°S. The planetary albedo over sea ice is

*α*

_{s}

*α*

_{W}

*γ*

*γ,*

*γ*= ln(100

*h*)/ln(100) for

*h*< 1 m and otherwise

*γ*= 1. The logarithmic dependence is based on the work of Ebert and Curry (1993).

Definitions of variables for the analytic model

^{1}

We plot *G* and *M* as functions of the annual mean thickness, while Untersteiner (1961) and Maykut (1986) plot them against the thickness at the beginning of the season.

^{2}

The net outflow for a region is the divergence of the volume flux: *O* = ∇ · *h***v** averaged over the region, where **v** is the ice velocity. If the wind and ocean stresses are independent of sea ice thickness and we consider the ice to be in free drift, we can assume **v** is nearly independent of *h*; then *O* is roughly proportional to *h.*