a. Representation of model state
模型状态的表示

SCMs typically compute the seasonally varying sea ice thickness h(t) as a prognostic variable. The surface temperature of the ice T is calculated diagnostically based on h and the surface energy flux. When the ice thickness reaches zero, SCMs often evolve the temperature of an ocean mixed layer of fixed depth (Thorndike 1992, and later studies). This temperature is taken to be vertically uniform in the ocean mixed layer and hence equal to the surface temperature T. To succinctly account for both ice thickness when ice is present and the temperature of the ocean mixed layer when ice is not present, an SCM can evolve the surface enthalpy E(t), which is defined as (Eisenman and Wettlaufer 2009)
SCM 通常将季节性变化的海冰厚度 h(t) 计算为预报变量。冰的表面温度 T 是基于 h 和表面能量通量进行诊断计算的。当冰厚度达到零时，SCM 通常会演变为固定深度的海洋混合层的温度（Thorndike 1992 及后续研究）。该温度在海洋混合层中被视为垂直均匀，因此等于表面温度 T。为了简洁地考虑冰存在时的冰厚度和冰不存在时的海洋混合层温度，SCM 可以演变表面焓 E(t)，其定义为（Eisenman 和 Wettlaufer 2009）。

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where L_f is the latent heat of fusion of sea ice and c_w is the heat capacity of the ocean mixed layer, which is equal to the product of the ocean mixed layer specific heat capacity, density, and depth. The melting point T_m is approximated to be constant at its freshwater value. The default values for all model parameters are discussed in section 2d and summarized in Table 1.
其中 L 是海冰的潜热，c _w 是海洋混合层的热容，等于海洋混合层比热容、密度和深度的乘积。熔点 T _m 被近似为其淡水值的常数。所有模型参数的默认值在第 2d 节中讨论，并在表 1 中总结。

Table 1. 表 1。

Default parameter values.
默认参数值。

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In contrast to SCMs, standard EBMs compute the equilibrium value of the spatially varying surface temperature T(x). Here, x ≡ sinθ with latitude θ, such that x = 0 at the equator and x = 1 at the North Pole.
与标准的 EBM 相比，SCM 计算空间变化的表面温度 T(x)的平衡值。在这里，x ≡ sinθ，其中θ为纬度，因此在赤道时 x = 0，在北极时 x = 1。

The model developed here evolves the surface enthalpy as a function of both time and latitude E(t, x). A standard SCM is represented in each latitudinal grid box, and the grid boxes are coupled via latitudinal diffusion of T, as is done in standard EBMs. The temperature T represents either (i) the temperature of the ocean mixed layer in the absence of ice or (ii) the ice surface temperature when ice is present. Below, we begin by discussing the seasonally varying EBM formulation that will be used to give spatial dependence, and then we add an SCM representation at each latitudinal grid box to evolve sea ice thickness. The reader who is primarily interested in the model results may skip to section 3.
这里开发的模型将表面焓作为时间和纬度 E(t, x) 的函数进行演变。每个纬度网格箱中都表示一个标准的 SCM，并且通过 T 的纬度扩散将网格箱耦合，正如标准的 EBM 所做的那样。温度 T 代表 (i) 在没有冰的情况下海洋混合层的温度，或 (ii) 在存在冰的情况下冰面温度。下面，我们首先讨论将用于提供空间依赖性的季节性变化 EBM 公式，然后在每个纬度网格箱中添加一个 SCM 表示，以演变海冰厚度。主要对模型结果感兴趣的读者可以跳到第 3 节。

b. EBM formulation with seasonal variations
b. 具有季节变化的 EBM 配方

The time evolution of E(t, x) is determined at each latitude by the net energy flux into the atmospheric column and surface below:
E(t, x)的时间演变在每个纬度上由进入大气柱和下面表面的净能量通量决定：

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which includes fluxes from top-of-atmosphere net solar radiation aS, outgoing longwave radiation (OLR) L, meridional heat transport in the atmosphere D∇²T, and heat flux into the model domain from the ocean below F_b (Fig. 1). Each of these terms is discussed below. Last, F in Eq. (2) represents a spatially uniform, seasonally constant climate forcing that is varied during some model runs. Increasing F can be interpreted as an increase in atmospheric CO₂, for example, analogous to global warming scenarios in GCM simulations.
包括来自大气顶层的净太阳辐射通量 aS、外辐射长波辐射 (OLR) L、大气中的经向热输送 D∇ ² T，以及来自下方海洋进入模型域的热通量 F _b （见图 1）。下面将讨论这些项。最后，方程 (2) 中的 F 代表一个空间均匀、季节性恒定的气候强迫，在某些模型运行中有所变化。增加 F 可以被解释为大气中 CO ₂ 的增加，例如，类似于全球气候模型 (GCM) 模拟中的全球变暖情景。

Note that most EBMs compute only the equilibrium climate state, ∂E/∂t = 0 in Eq. (2), whereas here we consider the time evolution of the system. Since we are considering an aquaplanet, and the ocean mixed layer has an effective heat capacity that is more than an order of magnitude higher than the atmosphere (North and Coakley 1979), we neglect the heat capacity of the atmospheric column. This implies that the vertical temperature profile in the atmosphere at a given time and latitude is fully determined by T.
请注意，大多数 EBM 仅计算平衡气候状态，即在方程（2）中∂E/∂t = 0，而在这里我们考虑系统的时间演变。由于我们考虑的是一个水星球，并且海洋混合层的有效热容比大气高出一个数量级（North 和 Coakley 1979），我们忽略了大气柱的热容。这意味着在给定时间和纬度下，大气中的垂直温度分布完全由 T 决定。

1) Solar radiation 1) 太阳辐射

Following North and Coakley (1979), we use an approximate representation of the insolation,
根据 North 和 Coakley（1979）的研究，我们使用辐射强度的近似表示

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where S₀ is the annual-mean insolation at the equator, S₁ determines the amplitude of seasonal insolation variations, ω = 2π yr⁻¹ is the annual frequency, and S₂ determines the equator-to-pole insolation gradient (Table 1). Note that the spatial dependence of S(t, x) is written here without using Legendre polynomials, which have often been used in the EBM literature as they enable analytical solutions. We choose this equivalent simpler representation because the model presented here is solved numerically.
其中 S ₀ 是赤道的年均辐射量，S ₁ 决定季节性辐射变化的幅度，ω = 2π yr ⁻¹ 是年频率，S ₂ 决定赤道到极地的辐射梯度（表 1）。请注意，S(t, x) 的空间依赖性在这里没有使用勒让德多项式，后者在 EBM 文献中经常被使用，因为它们能够提供解析解。我们选择这种等效的简单表示法，因为这里所呈现的模型是通过数值方法求解的。

Some of the insolation is reflected back to space and the rest is absorbed. The fraction of incident solar radiation that is absorbed is called the planetary, or top-of-atmosphere (TOA), coalbedo (i.e., one minus albedo). It depends on factors including the solar zenith angle, clouds, and the presence of reflective ice at the surface (Lian and Cess 1977). Similar to North (1975b), we write the planetary coalbedo as
一些入射辐射被反射回太空，其余部分被吸收。被吸收的入射太阳辐射的比例称为行星反照率或大气顶反照率（TOA），即一减去反照率。它依赖于包括太阳天顶角、云层以及表面反射冰的存在等因素（Lian 和 Cess 1977）。与 North（1975b）类似，我们将行星反照率表示为

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where a₀, a₂, and a_i are empirical parameters (see Table 1). It is the discrete jump in a(x, E) at the ice edge that introduces nonlinearity and therefore the possibility of multiple states. The addition of ice thickness evolution in the next section, however, will add further nonlinearity. Some studies have considered various smoothed albedo transitions between ice and water (e.g., Cahalan and North 1979; Eisenman and Wettlaufer 2009), which we eschew here for simplicity.
在这里， ₀ 、 ₂ 和一个是经验参数（见表 1）。在冰缘处，a(x, E)的离散跳跃引入了非线性，因此可能出现多种状态。然而，在下一节中加入冰厚度演变将进一步增加非线性。一些研究考虑了冰与水之间的各种平滑反照率过渡（例如，Cahalan 和 North 1979；Eisenman 和 Wettlaufer 2009），但为了简化，我们在此避免讨论。

2) OLR

Following Budyko (1969), we represent the OLR as a linear function of the surface temperature,
根据 Budyko（1969）的研究，我们将地表温度的外向长波辐射（OLR）表示为线性函数

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where A and B are empirical parameters (Table 1).
其中 A 和 B 是经验参数（表 1）。

3) Heat transport 热传输

Following North (1975b), the meridional heat flux in the climate system is approximated to be proportional to the meridional gradient of the surface temperature, thereby influencing the surface temperature evolution like a diffusive process with constant diffusivity D. Because of converging meridians on a spherical earth, meridional diffusion takes the form (North 1975b)
根据 North（1975b）的观点，气候系统中的经向热通量被近似为与表面温度的经向梯度成正比，从而像具有恒定扩散率 D 的扩散过程一样影响表面温度的演变。由于在球形地球上经线的汇聚，经向扩散呈现出以下形式（North 1975b）。

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The seasonal cycle introduces a small seasonally varying heat transport across the equator. For both conceptual transparency and computational convenience, we approximate this cross-equatorial heat flux to be zero, allowing us to restrict the model domain to a single hemisphere. Neglecting this in the EBM described in this section causes the temperature to change by no more than 0.3 K at any latitude at any time of year. Inserting Eq. (1) with E > 0 (open water) into Eq. (6) at x = 0 gives the equatorial boundary condition
季节循环引入了一个小的季节性变化的赤道热传输。为了概念上的透明性和计算的便利性，我们将这种赤道跨越热通量近似为零，从而使我们能够将模型域限制在单个半球。忽略这一点在本节描述的能量平衡模型中导致任何纬度在任何季节的温度变化不超过 0.3 K。在 x = 0 处将 E > 0（开水）代入方程（1）到方程（6）中，得到赤道边界条件。

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At the pole (x = 1), the heat transport in Eq. (6) goes to zero for any distribution of T because of converging meridians, and no boundary condition is necessary.
在极点（x = 1），由于经线汇聚，方程（6）中的热传输对于任何温度分布都趋于零，因此不需要边界条件。

c. Addition of SCM physics
c. SCM 物理的添加

Next, we add a representation of sea ice thickness evolution to the model and compute the resulting effect on the surface temperature. This is done by coupling Eq. (2) to an SCM representation of thermodynamic sea ice growth and melt (Fig. 1). Note that energy fluxes are represented in the present model at the top of the atmosphere, as in typical EBMs, in contrast to typical SCMs, which consider surface fluxes.
接下来，我们将海冰厚度演变的表征添加到模型中，并计算其对表面温度的影响。这是通过将方程（2）与热力学海冰生长和融化的 SCM 表征耦合来实现的（图 1）。请注意，当前模型中的能量通量是在大气顶部表示的，类似于典型的能量平衡模型（EBMs），而与典型的 SCMs 相反，后者考虑的是表面通量。

We approximately follow the SCM developed in Eisenman (2012). This SCM can be directly derived (cf. Eisenman and Wettlaufer 2009) as an approximate representation of the SCM of Maykut and Untersteiner (1971), which represents processes including vertical energy flux between the ice and the atmosphere, vertical conduction of heat within the ice, vertical heat flux between the ice and the ocean below (F_b), basal congelation during the winter growth season, and surface and basal ablation during the summer season.
我们大致遵循 Eisenman（2012）中开发的 SCM。该 SCM 可以直接推导（参见 Eisenman 和 Wettlaufer 2009），作为 Maykut 和 Untersteiner（1971）SCM 的近似表示，表示包括冰与大气之间的垂直能量通量、冰内部的垂直热传导、冰与下面海洋之间的垂直热通量（F _b ）、冬季生长季节的基底凝结，以及夏季的表面和基底消融等过程。

Vertical heat conduction within the ice is governed by a diffusion equation for the internal ice temperature. Here we assume that the energy associated with phase changes is much larger than the energy associated with temperature changes within the ice. Equivalently, the relevant Stefan number N_S ≡ (L_fΔh)(c_ihΔT)⁻¹ (Eisenman 2012), satisfies N_S ≫ 1, where c_i is the specific heat of the ice. In this approximation, the vertical diffusion gives a linear temperature profile, with temperature varying from T ≤ T_m at the top surface to T_m at the bottom surface. The heat flux upward through the ice is then equal to k(T_m − T)/h, where k is the ice thermal conductivity, which we treat as constant.
冰内的垂直热传导由内部冰温度的扩散方程控制。在这里，我们假设与相变相关的能量远大于与冰内温度变化相关的能量。等效地，相关的斯特凡数 N _S ≡ (LΔh)(chΔT) ⁻¹ (Eisenman 2012) 满足 N _S ≫ 1，其中 c 是冰的比热。在这个近似中，垂直扩散给出了线性温度分布，温度从顶部表面的 T ≤ T _m 变化到底部表面的 T _m 。通过冰向上的热流量等于 k(T _m − T)/h，其中 k 是冰的热导率，我们将其视为常数。

If the ocean mixed layer cools to T_m, ice begins to grow. As long as ice is present, the temperature of the ocean mixed layer is set to remain constant at T_m. In this regime, the ocean heat flux F_b goes directly into the bottom of the ice without losing heat to the ocean mixed layer.
如果海洋混合层降温至 T _m ，冰开始形成。只要冰存在，海洋混合层的温度将保持在 T _m 。在这种情况下，海洋热通量 F _b 直接传递到冰底部，而不会向海洋混合层散失热量。

Hence, when ice is present, the change in energy per unit area is equal to the change in E, and the ice thickness (h = −E/L_f) evolves according to Eq. (2). In other words, Eq. (2) applies equivalently for both ice-covered and ice-free conditions. In the model results, we identify the ice edge latitude θ_i as the latitude where E changes sign, and we similarly define x_i ≡ sinθ_i.
因此，当冰存在时，单位面积的能量变化等于 E 的变化，冰厚度（h = -E/L）根据公式（2）演变。换句话说，公式（2）在冰覆盖和无冰条件下同样适用。在模型结果中，我们将冰缘纬度θ定义为 E 变化符号的纬度，并同样定义 x ≡ sinθ。

There are two possible regimes when ice is present, separated by the surface melt threshold: either (i) T < T_m, in which case the net surface flux must be zero, or (ii) T = T_m, in which case surface melt occurs proportionally to the net surface flux. We determine which case applies by computing the surface temperature T₀ that would balance the surface energy flux and determining whether it is above or below the melting point. Since we treat the heat capacity of the atmosphere as negligible compared with the ocean mixed layer, the net surface energy flux is equal to the top-of-atmosphere energy flux plus the horizontal energy convergence in the atmosphere. Then T₀ is computed from
当冰存在时，有两种可能的状态，由表面融化阈值分隔：要么 (i) T < T _m ，在这种情况下，净表面通量必须为零；要么 (ii) T = T _m ，在这种情况下，表面融化与净表面通量成比例发生。我们通过计算表面温度 T ₀ 来确定适用的情况，该温度将平衡表面能量通量，并确定它是否高于或低于融化点。由于我们将大气的热容视为与海洋混合层相比可以忽略不计，净表面能量通量等于大气顶层能量通量加上大气中的水平能量汇聚。然后从以下公式计算 T ₀ ：

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If T₀ < T_m, then the surface flux is balanced by a subfreezing surface temperature, and the frozen regime (i) applies, with T = T₀. On the other hand, T₀ > T_m indicates that the surface flux cannot be balanced by a subfreezing surface temperature, and surface melt (ii) occurs. In this case the surface temperature remains at the melting point, T = T_m, and the surface melt is calculated using a Stefan condition, which states that the ice thinning from surface melt is equal to the net surface flux divided by L_f. Note that here the value of T₀ at a given location depends on T at other locations because of the ∇²T term, making this model considerably less straightforward to solve than a typical SCM.
如果 T ₀ < T _m ，则表面通量由低于冰点的表面温度平衡，适用冻结状态 (i)，此时 T = T ₀ 。另一方面，T ₀ > T _m 表明表面通量无法由低于冰点的表面温度平衡，发生表面融化 (ii)。在这种情况下，表面温度保持在熔点 T = T _m ，表面融化的计算使用 Stefan 条件，该条件指出，表面融化导致的冰层变薄等于净表面通量除以 L。请注意，这里某一位置的 T ₀ 值依赖于其他位置的 T 值，因为存在 ∇ ² T 项，使得该模型的求解比典型的 SCM 要复杂得多。

Ice-free surface conditions can be concisely included by adding a third regime for the surface temperature:
无冰表面条件可以通过为表面温度添加第三种状态来简洁地包含

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Equations (2), (8), and (9) determine the model, which represents the meridionally and seasonally varying global surface temperature and allows sea ice of varying thickness to form when the ocean mixed layer temperature reaches the melting point.
方程（2）、（8）和（9）确定了模型，该模型表示经度和季节变化的全球表面温度，并允许在海洋混合层温度达到融化点时形成不同厚度的海冰。

d. Default parameter values
d. 默认参数值

The default values of all model parameters are given in Table 1. Values for D, B, and c_w are adopted from North and Coakley (1979), where c_w corresponds to a 75-m ocean mixed layer depth. The values of D typically used in the literature range from D = 0.4 (Lin and North 1990) to 0.66 W m⁻² K⁻¹ (Rose and Marshall 2009). The insolation parameters S₀ and S₂ are adopted from the Legendre polynomial coefficients in North and Coakley (1979). Here we set the parameter dictating the seasonal amplitude of the solar forcing S₁ to be 25% larger than the astronomically based value used in North and Coakley (1979) in order to better match the observed seasonal range of sea ice thickness in the central Arctic. Note that this increase in S₁ may help account for factors such as seasonally varying cloud cover and water vapor that are not included in the model. Since North and Coakley (1979) do not consider an explicit albedo jump, we adopt coalbedo parameter values (a₀, a₂, and a_i) from North (1975b), converting the Legendre polynomial coefficients into our formalism.
所有模型参数的默认值见表 1。D、B 和 c _w 的值采用自 North 和 Coakley（1979），其中 c _w 对应于 75 米的海洋混合层深度。文献中通常使用的 D 值范围为 D = 0.4（Lin 和 North 1990）到 0.66 W m ⁻² K ⁻¹ （Rose 和 Marshall 2009）。日照参数 S ₀ 和 S ₂ 采用自 North 和 Coakley（1979）中的勒让德多项式系数。这里我们将决定太阳强迫季节振幅的参数 S ₁ 设置为比 North 和 Coakley（1979）中使用的天文基础值大 25%，以更好地匹配观察到的中央北极海冰厚度的季节范围。请注意，S ₁ 的增加可能有助于解释模型中未包含的季节性变化的云覆盖和水蒸气等因素。由于 North 和 Coakley（1979）未考虑显式的反照率跃迁，我们采用自 North（1975b）的煤反照率参数值（a ₀ 、a ₂ 和 a），将勒让德多项式系数转换为我们的形式。

For the heat flux upward from the ocean F_b, Maykut and Untersteiner (1971) use a value of F_b = 2 W m⁻² based on central Arctic observations. Thorndike (1992) adopts a possible range for F_b of 0–10 W m⁻². Other observations suggest an annual-mean value of F_b = 5 W m⁻² (Maykut and McPhee 1995). We adopt a value within the range of these previous estimates, F_b = 4 W m⁻².
对于来自海洋的向上热通量 F _b ，Maykut 和 Untersteiner (1971) 基于中央北极观测使用了 F _b = 2 W m ⁻² 的值。Thorndike (1992) 采用了 F _b 的可能范围为 0–10 W m ⁻² 。其他观测结果表明年均值 F _b = 5 W m ⁻² (Maykut 和 McPhee 1995)。我们采用了这些先前估计范围内的一个值，F _b = 4 W m ⁻² 。

We use values for the thermal conductivity k and latent heat of fusion L_f corresponding to pure ice.
我们使用与纯冰相对应的热导率 k 和熔化潜热 L 的值。

Last, we let A = 193 W m⁻², similar to the value in Mengel et al. (1988), in order to simulate an annual-mean hemispheric-mean temperature consistent with the observed present-day climate when F = 0.
最后，我们设定 A = 193 W m ⁻² ，与 Mengel 等人（1988）中的值相似，以便在 F = 0 时模拟与观察到的现代气候一致的年均半球均温。

In the model simulations presented in this study, we vary the climate forcing F, as well as D and S₁. For notational convenience, we denote the default values of the latter two parameters as D* and (Table 1).
在本研究中呈现的模型模拟中，我们改变了气候强迫 F，以及 D 和 S ₁ 。为了方便记号，我们将后两个参数的默认值表示为 D*和 （表 1）。

e. Numerical solution 数值解法

Numerically solving the system of Eqs. (2), (8), and (9) is not straightforward because of the implicit representation of T₀ in Eqs. (8) and (9). This represents a nonlinear ordinary differential equation in x that involves a free boundary between melting and freezing ice surfaces that would need to be solved at each time step in a standard forward Euler time-stepping routine. Furthermore, using forward Euler time-stepping for this diffusive system would require very high temporal resolution for numerical stability.
数值求解方程 (2)、(8) 和 (9) 的系统并不简单，因为方程 (8) 和 (9) 中 T ₀ 的隐式表示。这表示一个关于 x 的非线性常微分方程，涉及融化和冻结冰面之间的自由边界，需要在标准的前向欧拉时间步进过程中在每个时间步上进行求解。此外，对于这个扩散系统，使用前向欧拉时间步进需要非常高的时间分辨率以确保数值稳定性。

To efficiently integrate the model, we instead employ a numerical approach that solves a system of equations that can be visualized in terms of two separate layers. Horizontal diffusion occurs in a “ghost layer” with heat capacity c_g, all other processes occur in the main layer, and the temperature of the ghost layer is relaxed toward the temperature of the main layer with time scale τ_g. This method bypasses the need to solve a differential equation in x at each time step and allows us to use an implicit time-stepping routine with coarse time resolution in the ghost layer. We emphasize that the ghost layer does not represent a separate physical layer such as the atmosphere, which would add physical complexity to the model. This “two layer” system reduces to the system of Eqs. (2), (8), and (9) above in the limit that c_g → 0 and τ_g → 0. In the model simulations, c_g and τ_g are chosen to be sufficiently small that further reducing their values does not substantially influence the numerical solution (values given in Table 1). Further details are given in appendix A.
为了有效地整合模型，我们采用了一种数值方法，解决可以用两个独立层可视化的方程组。水平扩散发生在一个“幽灵层”中，其热容量为 c _g ，所有其他过程发生在主层中，幽灵层的温度随着时间尺度 τ _g 向主层的温度放松。该方法绕过了在每个时间步长中在 x 方向上求解微分方程的需要，并允许我们在幽灵层中使用隐式时间步进例程，具有粗略的时间分辨率。我们强调，幽灵层并不代表一个独立的物理层，例如大气层，这会给模型增加物理复杂性。这个“两层”系统在 c _g → 0 和 τ _g → 0 的极限下简化为上述方程 (2)、(8) 和 (9) 的系统。在模型模拟中，c _g 和 τ _g 被选择为足够小，以至于进一步减小它们的值不会对数值解产生实质性影响（值见表 1）。更多细节见附录 A。

The model is solved numerically in 0 ≤ x ≤ 1 using 400 meridional grid boxes that are equally spaced in x and 1000 time steps per year. The spatial resolution is required to sufficiently resolve the evolution of the sea ice cover, especially in parameter regimes where the SICI region is small. The temporal resolution is chosen to ensure numerical stability of the solution.
该模型在 0 ≤ x ≤ 1 的范围内通过 400 个在 x 方向上均匀分布的经度网格箱和每年 1000 个时间步长进行数值求解。空间分辨率要求足以解析海冰覆盖的演变，特别是在 SICI 区域较小的参数范围内。时间分辨率的选择旨在确保解的数值稳定性。

Code to numerically solve the present model, as well as the standard EBM described in section 2b but with S₁ = 0, is available online at http://eisenman.ucsd.edu/code.html.
用于数值求解当前模型的代码，以及在第 2b 节中描述的标准 EBM（但 S ₁ = 0），可在线获取，网址为 http://eisenman.ucsd.edu/code.html。

a. Simulated modern climate (F = 0)
模拟现代气候 (F = 0)

The simulated climate with all parameters at their default values (Table 1) and F = 0 is illustrated in Fig. 2, where the seasonal cycle of the equilibrated climate is plotted. Figure 2a shows the seasonal cycle of E(t, x), which fully represents the model state since E is the only prognostic variable and the forcing varies seasonally. The associated surface temperature (Fig. 2b) and ice thickness (Fig. 2c) are roughly consistent with present-day climate observations in the Northern Hemisphere. The simulated surface temperature varies approximately parabolic with x from an equatorial temperature of about 30°C throughout the year to a polar temperature of −10°C in winter and −3°C in summer (Fig. 2d). There is perennial sea ice around the pole (Fig. 2e) with ice thickness at the pole varying seasonally between 3.1 and 3.4 m (Fig. 2f). The edge of the sea ice cover ranges from θ_i = 58° in winter to 76° in summer (Fig. 2g), which can be compared with the observed zonal-mean ice edge latitude in the Northern Hemisphere (Eisenman 2010).
模拟气候在所有参数为默认值（表 1）且 F = 0 的情况下如图 2 所示，其中绘制了平衡气候的季节循环。图 2a 显示了 E(t, x)的季节循环，完全代表了模型状态，因为 E 是唯一的预报变量，强迫因子随季节变化。相关的表面温度（图 2b）和冰厚（图 2c）与北半球当前的气候观测大致一致。模拟的表面温度在 x 方向上大致呈抛物线变化，从赤道温度约 30°C 到冬季极地温度−10°C 和夏季−3°C（图 2d）。极地周围存在常年海冰（图 2e），极地的冰厚在季节间变化，介于 3.1 和 3.4 米之间（图 2f）。海冰覆盖的边缘在冬季为θ = 58°，夏季为 76°（图 2g），这可以与北半球观测到的纬向平均冰缘纬度进行比较（Eisenman 2010）。

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Simulated climate in the default parameter regime. Contour plot of the seasonal cycle of (a) surface enthalpy E(x, t), (b) surface temperature T(x, t), and (c) sea ice thickness h(x, t). The black curve in (a)–(c) indicates the ice edge. (d) Surface temperature T in summer and winter, corresponding to dashed and solid vertical lines in (c). (e) Ice thickness h in summer and winter where x > 0.7. (f) Seasonal cycle of ice thickness at the pole h_p. (g) Seasonal cycle of the latitude of the sea ice edge θ_i.
默认参数范围内的模拟气候。季节循环的等高线图 (a) 表面焓 E(x, t)，(b) 表面温度 T(x, t)，和 (c) 海冰厚度 h(x, t)。图 (a)–(c) 中的黑色曲线表示冰缘。(d) 夏季和冬季的表面温度 T，对应于 (c) 中的虚线和实线垂直线。(e) 夏季和冬季的冰厚度 h，其中 x > 0.7。(f) 极地冰厚度的季节循环 h _p 。(g) 海冰缘纬度的季节循环 θ。

Citation: Journal of Climate 28, 10; 10.1175/JCLI-D-14-00654.1
引用：气候学杂志 28, 10; 10.1175/JCLI-D-14-00654.1

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The presence of a substantial seasonal sea ice cover in the simulated climate (Fig. 2c), which is qualitatively consistent with observations, stands in contrast with typical SCM results. It has been a conundrum in SCM studies that they have often struggled to simulate a stable seasonal sea ice cover. The SCM in Thorndike (1992) was forced by a square-wave seasonal cycle, with insolation and longwave radiation switching between a constant summer value for half the year and a constant winter value for the other half. It did not feature any stable seasonal ice. The SCM in Eisenman and Wettlaufer (2009), which had smoothly varying insolation and longwave radiation, did feature stable seasonal ice, but only in a relatively small range of climate forcings. In Eisenman and Wettlaufer (2009, see their Fig. S5), this stable seasonal ice was hypothesized to arise because of a larger seasonal amplitude in the longwave forcing than was used by Thorndike (1992). Moon and Wettlaufer (2012) proposed the contrasting hypothesis that the square-wave seasonal cycle in solar forcing was the essential factor preventing the Thorndike (1992) SCM from simulating a stable seasonal ice cover. They suggested that resolving the seasonal cycle in solar forcing with at least two different values during summer was necessary for stable seasonal ice.
在模拟气候中，存在显著的季节性海冰覆盖（图 2c），其定性与观测结果一致，这与典型的单层气候模型（SCM）结果形成对比。在 SCM 研究中，稳定的季节性海冰覆盖的模拟一直是一个难题。Thorndike（1992）中的 SCM 是通过方波季节循环强迫的，日照和长波辐射在一年中的一半时间内切换为恒定的夏季值，而在另一半时间内切换为恒定的冬季值。该模型没有表现出任何稳定的季节性冰。Eisenman 和 Wettlaufer（2009）中的 SCM 具有平滑变化的日照和长波辐射，确实表现出稳定的季节性冰，但仅在相对较小的气候强迫范围内。在 Eisenman 和 Wettlaufer（2009，见其图 S5）中，假设这种稳定的季节性冰是由于长波强迫的季节振幅大于 Thorndike（1992）所使用的值。Moon 和 Wettlaufer（2012）提出了相反的假设，认为太阳强迫中的方波季节循环是阻止 Thorndike（1992）SCM 模拟稳定季节性冰覆盖的关键因素。 他们建议，在夏季使用至少两个不同的值来解决太阳强迫的季节循环是实现稳定季节性冰层所必需的。

The results presented here suggest an alternative resolution, namely that the inclusion of a horizontal dimension is an essential ingredient for accurately simulating a stable seasonal ice cover. In contrast with Moon and Wettlaufer (2012), who found that square-wave solar forcing did not allow seasonal ice, the model developed here still simulates a large seasonal migration of the ice edge when the cos ωt factor in Eq. (3) is replaced with a square wave (not shown). In contrast with Eisenman and Wettlaufer (2009), who found a narrow SCM forcing range that allowed for stable seasonal ice, seasonal ice occurs in this spatially varied model over a wide range of latitudes.
这里呈现的结果表明了一种替代解决方案，即引入一个横向维度是准确模拟稳定季节性冰盖的一个重要因素。与 Moon 和 Wettlaufer（2012）发现的方波太阳强迫不允许季节性冰的结论相反，本文开发的模型在将方程（3）中的 cos ωt 因子替换为方波时，仍然模拟出冰缘的大规模季节性迁移（未显示）。与 Eisenman 和 Wettlaufer（2009）发现的允许稳定季节性冰的狭窄 SCM 强迫范围相反，在这个空间变化的模型中，季节性冰在广泛的纬度范围内出现。

b. Forced warming (increasing F)
强制加热（增加 F）

Next, we numerically simulate the equilibrium response of the model to changes in the climate forcing by slowly ramping up F. Beginning with a 200-yr spinup simulation at the initial forcing value (F = −10 W m⁻²), we increase F in increments of 0.2 W m⁻² and integrate the system for 40 years at each value of F. This approach is used in all of the following simulations of the model response to increasing or decreasing climate forcing.
接下来，我们通过缓慢增加 F 来数值模拟模型对气候强迫变化的平衡响应。从初始强迫值（F = -10 W m ⁻² ）开始进行 200 年的预热模拟，我们以 0.2 W m ⁻² 的增量增加 F，并在每个 F 值下集成系统 40 年。这种方法在以下所有模型对气候强迫增加或减少的响应模拟中均被使用。

As shown in Fig. 3, the climate steadily warms and the seasonally varying sea ice cover steadily recedes until the pole is ice free throughout the year. The summer ice disappears at F = 2.5 W m⁻² and the winter ice at F = 11 W m⁻² (see also Fig. 4a).
如图 3 所示，气候持续变暖，季节性变化的海冰覆盖稳步减少，直到极地全年无冰。夏季冰在 F = 2.5 W m ⁻² 时消失，冬季冰在 F = 11 W m ⁻² 时消失（另见图 4a）。

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Simulated warming and sea ice loss in the default parameter regime with increasing climate forcing, F. Shown are the annual-mean (a) surface temperature T, (b) surface enthalpy E, and (c) ice thickness h, as functions of location x and forcing F. Black dashed lines show the summer (above) and winter (below) locations of the sea ice edge.
在默认参数范围内，随着气候强迫的增加，模拟的变暖和海冰损失。显示的是年均（a）表面温度 T，（b）表面焓 E，以及（c）冰厚度 h，作为位置 x 和强迫 F 的函数。黑色虚线显示了夏季（上方）和冬季（下方）海冰边缘的位置。

Citation: Journal of Climate 28, 10; 10.1175/JCLI-D-14-00654.1
引用：气候学杂志 28, 10; 10.1175/JCLI-D-14-00654.1

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Simulated sea ice loss and recovery in the default parameter regime with increasing and decreasing climate forcing, F. Shown are the ice area A_i vs (a) the forcing F and (b) the annual-mean hemispheric-mean surface temperature anomaly. The solid thick red and thin blue lines represent the annual-mean ice area in the warming and cooling scenarios, respectively; note that they coincide, implying no hysteresis. The faint solid and dashed lines show the winter and summer ice areas in the warming scenario.
模拟海冰损失和恢复在默认参数范围内，随着气候强迫的增加和减少，F。图中显示了冰面积 A 与 (a) 强迫 F 和 (b) 年均半球平均表面温度异常的关系。粗红线和细蓝线分别代表升温和降温情景下的年均冰面积；注意它们重合，意味着没有滞后现象。淡实线和虚线显示了升温情景下的冬季和夏季冰面积。

Citation: Journal of Climate 28, 10; 10.1175/JCLI-D-14-00654.1
引用：气候学杂志 28, 10; 10.1175/JCLI-D-14-00654.1

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c. Test for hysteresis c. 滞后测试

It is noteworthy that the sea ice declines smoothly, with no jumps occurring during the transition from perennial sea ice to seasonally ice-free conditions and then to perennially ice-free conditions. Rather, the summer and winter sea ice edges both respond fairly linearly to F (dashed lines in Fig. 3). This can similarly be considered in terms of the ice area, given by A_i = A_H(1 − x_i), where A_H is the surface area of the hemisphere (Fig. 4a).
值得注意的是，海冰的减少是平滑的，在从多年冰到季节性无冰条件再到永久无冰条件的过渡过程中没有发生跳跃。相反，夏季和冬季海冰边缘对 F 的响应都相当线性（图 3 中的虚线）。从冰面积的角度来看，这也可以类似地考虑，冰面积由 A = A _H (1 − x)给出，其中 A _H 是半球的表面积（图 4a）。

Figure 4b shows that the loss of summer and winter ice also relates linearly to the annual-mean hemispheric-mean temperature. The model becomes seasonally ice free at 2°C of warming, and it becomes perennially ice free at 6°C (Fig. 4b). These values compare with comprehensive GCM results from Armour et al. (2011), who find the complete loss of summer and winter ice to occur at 4° and 7°C, respectively.
图 4b 显示，夏季和冬季冰的损失与年均半球平均温度也呈线性关系。模型在升温 2°C 时季节性无冰，而在升温 6°C 时则全年无冰（图 4b）。这些数值与 Armour 等人（2011）的综合 GCM 结果相比较，他们发现夏季和冬季冰的完全消失分别发生在 4°C 和 7°C。

After the climate has become perennially ice free, we slowly ramp F back down again. We find that the ice recovers during cooling along the same trajectory as the ice retreat during warming, with no hysteresis (Fig. 4). The linearity and reversibility of the response in the present model is consistent with results from most comprehensive GCMs (e.g., Winton 2006; Armour et al. 2011; Winton 2011), and it is in contrast with previous results from EBMs and SCMs.
在气候变得永久无冰后，我们慢慢将 F 降低。我们发现冰在降温过程中以与升温期间冰退缩相同的轨迹恢复，没有滞后现象（图 4）。当前模型中响应的线性和可逆性与大多数综合气候模型（例如，Winton 2006；Armour 等，2011；Winton 2011）的结果一致，这与之前的能量平衡模型和单层模型的结果形成对比。

a. EBM regime (S₁ = 0)
EBM 体制 (S ₁ = 0)

When the seasonal amplitude is set to zero in the present model (S₁ = 0), the steady-state solution is equivalent to the solution of a standard annual-mean EBM. In this limit, the equilibrium state of the system becomes time independent, with Eq. (2) reducing to
当当前模型中的季节幅度设置为零时（S ₁ = 0），稳态解等同于标准年均能量平衡模型的解。在这个极限情况下，系统的平衡状态变得与时间无关，方程（2）简化为

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Here, the time evolution of the enthalpy E is no longer included, so the surface temperature can be determined directly from this ordinary differential equation in x, without consideration of the vertical heat conduction within the ice.
在这里，焓 E 的时间演变不再被考虑，因此可以直接从这个关于 x 的常微分方程中确定表面温度，而无需考虑冰内的垂直热传导。

Equation (10) is equivalent to the formulation of a standard annual-mean EBM [e.g., Eq. (22) in North et al. (1981)]. Analytical solutions can be found by making use of the property that Legendre polynomials are eigenfunctions of the diffusion operator Eq. (6). They are, however, complicated by the dependence of the coalbedo on T. Following Held and Suarez (1974), we approximately solve Eq. (10) for F as a function of x_i [e.g., Eqs. (28), (29), and (37) in North et al. (1981)], rather than vice versa. The approximate analytical solution involves a sum of even Legendre polynomials, and we retain terms up to the 40th degree, similar to previous studies (e.g., Mengel et al. 1988).
方程（10）等同于标准年均能量平衡模型的公式 [例如，North 等人（1981）中的方程（22）]。通过利用勒让德多项式是扩散算子方程（6）的特征函数的性质，可以找到解析解。然而，煤反照率对温度的依赖性使得问题变得复杂。遵循 Held 和 Suarez（1974）的研究，我们将方程（10）近似求解为 F 关于 x 的函数 [例如，North 等人（1981）中的方程（28）、（29）和（37）]，而不是反过来。近似解析解涉及偶数勒让德多项式的和，我们保留高达 40 次的项，类似于之前的研究（例如，Mengel 等人 1988）。

Figure 5a shows both the analytical solution and numerical model results. It illustrates the evolution of the sea ice edge x_i under varied climate forcing F with S₁ = 0 and D = D*. The figure illustrates sea ice retreat in the numerical simulation as F is increased (red) until an ice-free state is reached, followed by the onset and advance of the sea ice cover when F is subsequently decreased (blue).
图 5a 显示了分析解和数值模型结果。它展示了在不同气候强迫 F 下，海冰边缘 x 的演变，其中 S ₁ = 0 且 D = D*。该图展示了在数值模拟中，随着 F 的增加（红色），海冰的退缩，直到达到无冰状态，随后当 F 再次降低时（蓝色），海冰覆盖的出现和扩展。

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(a) Steady-state locations of the simulated ice edge (x_i) under varied climate forcing (F) in the parameter regime where the model reduces to an EBM (D = D*, S₁ = 0). The numerical solution is indicated for F being slowly ramped up (red) and then back down (blue). Stable and unstable states from the analytic EBM solution are also shown as solid and dashed black lines, respectively. (b) As in (a), but in terms of the enthalpy at the pole (E_p) rather than the ice edge location. The vertical axis on the right indicates corresponding values of the surface temperature at the pole T_p (for E > 0; green) and ice thickness at the pole h_p (for E < 0; orange). Both panels also indicate ΔF that is defined as the difference between the bifurcation points F_c (blue square) and F_w (red square).
(a) 在模型简化为能量平衡模型（EBM）（D = D*，S ₁ = 0）的参数范围内，模拟冰缘（x）在不同气候强迫（F）下的稳态位置。数值解在 F 缓慢上升（红色）和再下降（蓝色）时的情况被指示。来自解析 EBM 解的稳定和不稳定状态分别以实线和虚线黑色线条表示。(b) 与(a)相同，但以极地的焓（E _p ）而非冰缘位置为依据。右侧的纵轴表示极地表面温度 T _p （对于 E > 0；绿色）和极地冰厚 h _p （对于 E < 0；橙色）的相应值。两个面板还指示了ΔF，定义为分岔点 F _c （蓝色方块）和 F _w （红色方块）之间的差异。

Citation: Journal of Climate 28, 10; 10.1175/JCLI-D-14-00654.1
引用：气候学杂志 28, 10; 10.1175/JCLI-D-14-00654.1

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The hysteresis loop corresponds to the SICI. Note that a second hysteresis loop corresponding to the snowball earth instability occurs in substantially colder climates, as in standard EBMs. Figure 5a highlights the close agreement between numerical results (blue/red) and the analytical solution (black). It can be shown that the climates indicated in Fig. 5 are stable where the slope is positive (solid black) and unstable where the slope is negative (dashed black) (North 1975a). The system therefore does not support an ice cover with an equilibrium ice edge poleward of x_i = 0.98, or 79° latitude.
滞后回路对应于 SICI。请注意，第二个滞后回路对应于雪球地球不稳定性，发生在显著更冷的气候中，如标准 EBM 所示。图 5a 突出了数值结果（蓝色/红色）与解析解（黑色）之间的密切一致性。可以证明，图 5 中所示的气候在斜率为正（实线黑色）时是稳定的，而在斜率为负（虚线黑色）时是不稳定的（North 1975a）。因此，该系统不支持在 x = 0.98 或 79°纬度以北的冰盖与平衡冰缘。

A measure of the extent of bistability in the system can be obtained by considering the width of the hysteresis loop. We define two critical values of the forcing F: (i) F_w is the value at which the system first transitions to a perennially ice-free pole in a warming scenario, and (ii) F_c is the value at which the wintertime ice cover first reappears in a cooling scenario. These two values of F are indicated in Fig. 5 by a red square and a blue square, respectively. A saddle-node bifurcation occurs at each of these values in the parameter regime shown in Fig. 5. The width of the hysteresis loop is then defined as
系统中双稳态程度的量度可以通过考虑滞回环的宽度来获得。我们定义了两个强迫 F 的临界值：(i) F _w 是系统在变暖情景下首次过渡到永久无冰极点的值，(ii) F _c 是系统在变冷情景下冬季冰盖首次重新出现的值。这两个 F 的值分别在图 5 中用红色方块和蓝色方块表示。在图 5 所示的参数范围内，这两个值处发生鞍结分岔。滞回环的宽度定义为

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Note that ΔF may be seen as a societally relevant measure of instability and associated irreversibility, since it indicates how much the radiative forcing would need to be reduced for the sea ice to return after crossing a tipping point during global warming, although it should be noted that this requires long time scales for the climate system to equilibrate.
请注意，ΔF 可以被视为一个与社会相关的不稳定性和相关不可逆性的度量，因为它表明在全球变暖过程中，海冰在越过临界点后需要减少多少辐射强迫才能恢复，尽管需要指出的是，这需要气候系统在较长时间尺度上达到平衡。

It is useful to also consider the hysteresis loop in terms of the enthalpy at the pole E_p. Computing F_c and F_w from E_p (as defined in Fig. 5b) gives equivalent results to using x_i (as defined in Fig. 5a), since the bifurcations are associated with a transition between an ice-covered pole and an ice-free pole. Here we use both methods interchangeably to compute the bifurcation points, for example using E_p in the numerical results when D = 0.
考虑极点 E _p 的焓的滞回回路也是有用的。从 E _p 计算 F _c 和 F _w （如图 5b 所定义）得到的结果与使用 x（如图 5a 所定义）相当，因为分岔与冰盖极和无冰极之间的转变相关。在这里，我们交替使用这两种方法来计算分岔点，例如在 D = 0 时在数值结果中使用 E _p 。

b. SCM regime (D = 0)
b. SCM 体制 (D = 0)

The present model reduces to an array of independent SCMs when meridional heat transport is turned off by setting D = 0. In this case no communication occurs between individual locations and the state of each grid point is determined by the balance of vertical heat fluxes alone. For a given value of x, Eq. (2) reduces to an ordinary differential equation in time representing a standard SCM [e.g., Eq. (2) in Eisenman (2012)]. As in typical SCMs, we find bistability in the parameter regime (D = 0, S₁ = ), with ΔF = 7.0 W m⁻².
当前模型在关闭经向热输送（设置 D = 0）时简化为一系列独立的气候模型（SCMs）。在这种情况下，各个位置之间没有通信，每个网格点的状态仅由垂直热通量的平衡决定。对于给定的 x 值，方程（2）简化为一个关于时间的常微分方程，表示一个标准的气候模型 [例如，Eisenman (2012) 中的方程（2）]。与典型的气候模型一样，我们在参数范围 (D = 0, S ₁ = ) 中发现了双稳态，ΔF = 7.0 W m ⁻² 。

a. Full (D, S₁) parameter space
完整的 (D, S ₁ ) 参数空间

To address this, we performed 441 model simulations that explore how the hysteresis width ΔF varies in the (D, S₁) parameter space. We use 21 evenly spaced values for D in the range [0, 0.76] W m⁻² K⁻¹ and 21 evenly spaced values for S₁ in the range [0, 351] W m⁻². In each simulation, F is ramped up and then down, and ΔF is computed. The results are shown in Fig. 6.
为了解决这个问题，我们进行了 441 次模型模拟，探讨滞后宽度ΔF 在(D, S ₁ )参数空间中的变化。我们在[0, 0.76] W m ⁻² K ⁻¹ 范围内使用 21 个均匀分布的 D 值，在[0, 351] W m ⁻² 范围内使用 21 个均匀分布的 S ₁ 值。在每次模拟中，F 先上升再下降，并计算ΔF。结果如图 6 所示。

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Dependence of the extent of bistability ΔF on horizontal communication D and seasonal amplitude S₁ summarizing the main result of this study. Parameter values associated with typical SCMs and EBMs are indicated, as well as the default parameter values in the present model.
双稳态程度ΔF 对水平通信 D 和季节振幅 S 的依赖性 ₁ 总结了本研究的主要结果。与典型的气候系统模型（SCMs）和能量平衡模型（EBMs）相关的参数值被指示，同时也给出了当前模型中的默认参数值。

Citation: Journal of Climate 28, 10; 10.1175/JCLI-D-14-00654.1
引用：气候学杂志 28, 10; 10.1175/JCLI-D-14-00654.1

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Figure 6 indicates how the inclusion of both spatial and seasonal variations influences the presence of bistability. Beginning in the EBM regime (D = 0, S₁ = ) and adding a diffusivity of just D = 0.1D* suffices to eliminate any hysteresis in the model. Beginning in the SCM regime (D = D*, S₁ = 0) and adding a seasonality of just S₁ = 0.2 similarly eliminates any hysteresis. Last, if both parameters are reduced by equal factors from their default values (D = D*, S₁ = ), bistability does not occur until (D, S₁) < 0.3(D*, ). In other words, including even a relatively small amount of both meridional and seasonal variations destroys the bistability that occurs in SCMs and EBMs.
图 6 显示了空间和季节变化的结合如何影响双稳态的存在。从 EBM 状态开始（D = 0，S ₁ = ），仅添加 D = 0.1D*的扩散性就足以消除模型中的任何滞后。从 SCM 状态开始（D = D*，S ₁ = 0），仅添加 S ₁ = 0.2 的季节性同样消除了任何滞后。最后，如果两个参数都从其默认值（D = D*，S ₁ = ）按相同的比例减少，双稳态直到(D, S ₁ ) < 0.3(D*, )时才会出现。换句话说，即使是相对较小的经向和季节变化的结合也会破坏 SCM 和 EBM 中发生的双稳态。

The results in Fig. 6 summarize the main finding of this study. In the following subsections we discuss the underlying mechanisms. We begin by considering the value of ΔF at the origin in Fig. 6, followed by the limiting cases along the horizontal axis (S₁ = 0) and vertical axis (D = 0).
图 6 中的结果总结了本研究的主要发现。在接下来的小节中，我们讨论潜在的机制。我们首先考虑图 6 中原点处的ΔF 值，然后讨论沿水平轴（S ₁ = 0）和垂直轴（D = 0）的极限情况。

b. No seasonal cycle and no heat transport (D = 0, S₁ = 0)
b. 无季节循环且无热量输送（D = 0，S ₁ = 0）

In the simplest version of the model there is no seasonal cycle (S₁ = 0) and no heat transport (D = 0). In this case, F_c is the forcing associated with the pole being ice free at T = T_m, and F_w is the forcing associated with the pole being ice covered at T = T_m. Hence the width of the hysteresis loop (ΔF) is equal to the jump in solar forcing between ice-covered and ice-free conditions at the pole,
在模型的最简单版本中，没有季节循环（S ₁ = 0）和热量传输（D = 0）。在这种情况下，F _c 是与极地在 T = T _m 时无冰状态相关的强迫，F _w 是与极地在 T = T _m 时有冰覆盖状态相关的强迫。因此，滞回环的宽度（ΔF）等于极地在有冰覆盖和无冰状态之间的太阳强迫的跳跃。

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as can be seen from Eqs. (2)–(4). It is interesting to note in Fig. 6 how the region of bistability connects this point (D = 0, S₁ = 0), the standard EBM regime (D = D*, S₁ = 0), and the standard SCM regime (D = 0, S₁ = ). This may be taken to imply that these three limiting cases of bistability all effectively represent the same physical process, namely multiple stable states due to the ice–albedo feedback.
如方程（2）至（4）所示。在图 6 中有趣的是，双稳态区域如何连接这个点（D = 0, S ₁ = 0）、标准的 EBM 状态（D = D*, S ₁ = 0）和标准的 SCM 状态（D = 0, S ₁ = ）。这可以被理解为这三种双稳态的极限情况实际上都有效地代表了相同的物理过程，即由于冰-反照率反馈导致的多重稳定状态。

c. Stabilization from horizontal transport (S₁ = 0)
c. 来自水平运输的稳定性 (S ₁ = 0)

Here we consider how ΔF varies along the horizontal axis of Fig. 6 (i.e., in the EBM regime). Polar temperatures are typically higher for larger meridional heat transport. As D increases, the sea ice cover therefore disappears and also reappears at lower values of F (i.e., F_w and F_c both decrease). This is illustrated in Fig. 7a, where the critical forcings, F_w and F_c, are plotted versus D for both the numerical solution and the analytical solution discussed in section 4a. For the range of forcings F < F_c, the model does not feature SICI bistability, with the only stable state being a finite ice cover. Similarly, the range F > F_w only allows stable states with an ice-free pole. In the range F_c < F < F_w both ice-covered and ice-free conditions are stable, resulting in SICI bistability. These three regimes are illustrated in the inset of Fig. 7a.
在这里，我们考虑ΔF 在图 6 的水平轴上如何变化（即在 EBM 状态下）。极地温度通常在较大的经向热输送下较高。随着 D 的增加，海冰覆盖因此消失，并且在较低的 F 值（即 F _w 和 F _c 均减少）时重新出现。这在图 7a 中得到了说明，其中临界强迫 F _w 和 F _c 相对于 D 的数值解和在 4a 节中讨论的解析解进行了绘制。在强迫范围 F < F _c 内，模型不具有 SICI 双稳态，唯一的稳定状态是有限的冰覆盖。同样，范围 F > F _w 仅允许无冰极的稳定状态。在范围 F _c < F < F _w 内，冰覆盖和无冰条件均为稳定，导致 SICI 双稳态。这三个状态在图 7a 的插图中得到了说明。

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Dependence of the critical forcing values on D in the EBM regime (S₁ = 0). (a) F_c (blue) and F_w (red) vs D for numerical results (squares) and analytical results (solid line). The inset indicates the bistable regime. (b) Hysteresis width ΔF vs D for numerical (squares) and analytical (solid line) approximate solutions. Also indicated are the default value D = D* (dashed vertical line) and the value at which the SICI merges with the snowball earth instability D = D_max (solid vertical line).
在 EBM 区域中，临界强迫值对 D 的依赖性（S ₁ = 0）。(a) F _c （蓝色）和 F _w （红色）与 D 的关系，数值结果（方块）和解析结果（实线）。插图表示双稳态区域。(b) 滞后宽度ΔF 与 D 的关系，数值（方块）和解析（实线）近似解。还标出了默认值 D = D*（虚线垂直线）和 SICI 与雪球地球不稳定性合并时的值 D = D _max （实线垂直线）。

Citation: Journal of Climate 28, 10; 10.1175/JCLI-D-14-00654.1
引用：气候学杂志 28, 10; 10.1175/JCLI-D-14-00654.1

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The dependence of ΔF on D is shown in Fig. 7b. Note that the numerical results approach the analytical results when the spatial resolution increases. We find that ΔF typically decreases with increasing D, with the greatest sensitivity of ΔF occurring when D is small. This indicates that the introduction of diffusive communication between different latitudes has a stabilizing effect on the polar sea ice cover.
ΔF 对 D 的依赖关系如图 7b 所示。注意，当空间分辨率增加时，数值结果接近于解析结果。我们发现，ΔF 通常随着 D 的增加而减小，ΔF 对 D 的敏感性在 D 较小时最为显著。这表明，不同纬度之间引入扩散通信对极地海冰覆盖具有稳定作用。

At a critical diffusivity D_max (marked by a solid vertical line in Fig. 7), the small ice cap instability merges with the snowball earth instability. For D > D_max, there is no stable climate with 0 < x_i < 1. This property of EBMs for large values of D has been noted previously (Lindzen and Farrell 1977). For these values of D the global surface temperature becomes relatively isothermal, allowing stable solutions only with T < T_m everywhere or T > T_m everywhere.
在临界扩散率 D _max （在图 7 中用实线垂直线标记）处，小冰盖不稳定性与雪球地球不稳定性合并。对于 D > D _max ，在 0 < x < 1 的情况下没有稳定气候。EBMs 在大值 D 下的这一特性之前已被注意到（Lindzen 和 Farrell 1977）。对于这些 D 值，全球表面温度变得相对等温，仅在 T < T _m 或 T > T _m 的情况下允许稳定解。

The stabilizing effect of increased diffusivity when D < D_max can be visualized by considering a potential V associated with the model evolution. We identify the model state by the surface temperature at the pole T_p and we define the potential such that dT_p/dt = −dV/dT_p. Hence valleys in V are stable equilibria and peaks in V are unstable equilibria. The computation of V is described in appendix C.
当 D < D _max 时，增加扩散率的稳定效应可以通过考虑与模型演化相关的势能 V 来可视化。我们通过极地的表面温度 T _p 来识别模型状态，并定义势能，使得 dT _p /dt = −dV/dT _p 。因此，V 的谷底是稳定的平衡点，而 V 的峰值是不稳定的平衡点。V 的计算在附录 C 中进行了描述。

We compare two potentials in Fig. 8. One has a smaller diffusivity (D = 0.12 W m⁻² K⁻¹) and the other has a larger diffusivity (D = 0.14 W m⁻² K⁻¹). We choose values of F such that the unstable equilibria occur at the same value of T_p, using F = 55 and 51 W m⁻², respectively. As one might expect intuitively from the effect of diffusion, the higher diffusivity is associated with a smoother potential. In other words, there is a smaller barrier between the two stable equilibria, which suggests that a smaller change in forcing is necessary to transition between the states (i.e., ΔF is reduced).
我们在图 8 中比较了两个势能。一个具有较小的扩散率（D = 0.12 W m ⁻² K ⁻¹ ），另一个具有较大的扩散率（D = 0.14 W m ⁻² K ⁻¹ ）。我们选择 F 的值，使得不稳定平衡在相同的 T _p 值下发生，分别使用 F = 55 和 51 W m ⁻² 。正如人们从扩散的影响中直观地预期的那样，较高的扩散率与更平滑的势能相关。换句话说，两个稳定平衡之间的障碍较小，这表明在状态之间过渡所需的强迫变化较小（即，ΔF 减少）。

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Potential (V) in the EBM regime (S₁ = 0) with D = 0.12 W m⁻² K⁻¹, F ≡ 55 W m⁻² (solid) and with D = 0.14 W m⁻² K⁻¹, F ≡ 51 W m⁻² (dashed), illustrating how increasing D reduces the barrier between stable equilibria by smoothing the potential wells.
在 EBM 区域（S ₁ = 0）中，D = 0.12 W m ⁻² K ⁻¹ 时，F ≡ 55 W m ⁻² （实线）和 D = 0.14 W m ⁻² K ⁻¹ 时，F ≡ 51 W m ⁻² （虚线），说明增加 D 如何通过平滑势阱来减少稳定平衡之间的障碍。

Citation: Journal of Climate 28, 10; 10.1175/JCLI-D-14-00654.1
引用：气候学杂志 28, 10; 10.1175/JCLI-D-14-00654.1

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Note that explanations involving the diffusive length scale associated with Eq. (10) have also been suggested for the influence of D on the extent of the SICI (Lindzen and Farrell 1977; North 1984). Furthermore, the qualitative relationship we find between D and ΔF is consistent with the conclusion in Eisenman (2012) that that parameter changes in climate models that give rise to thinner ice and warmer ice-free ocean surface temperatures make the system less prone to bistability.
请注意，涉及与方程（10）相关的扩散长度尺度的解释也被提出用于 D 对 SICI 范围影响的研究（Lindzen 和 Farrell 1977；North 1984）。此外，我们发现 D 与ΔF 之间的定性关系与 Eisenman（2012）中的结论一致，即气候模型中导致冰层变薄和无冰海洋表面温度升高的参数变化使系统更不容易出现双稳态。

d. Stabilization from seasonal cycle (D = 0)
d. 来自季节循环的稳定性 (D = 0)

The influence of the amplitude of the seasonal cycle (S₁) in the SCM regime (D = 0) is illustrated in Fig. 9. In this regime, F_c can be readily obtained analytically because it is the point of transition from a perennially ice-free state, in which case the model equations are linear. We find
季节循环幅度（S ₁ ）在 SCM 模式（D = 0）中的影响如图 9 所示。在该模式下，F _c 可以通过解析方法轻松获得，因为这是从永久无冰状态过渡的点，在这种情况下，模型方程是线性的。我们发现

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where κ ≡ [1 + (ωc_w/B)²]^−1/2 (blue line in Fig. 9a).
其中 κ ≡ [1 + (ωc _w /B) ² ] ^−1/2 (图 9a 中的蓝线)。

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Dependence of the critical forcing values on S₁ in the SCM regime (D = 0). (a) Numerical results for F_w (red squares) and F_c (blue squares) vs S₁ and the exact analytical solutions for F_c (blue line). The exact analytical solutions for F_w when ice thickness evolution is neglected is also included (red line). (b) Hysteresis width ΔF vs S₁ for numerical results (squares), as well as the analytical solution that neglects ice thickness evolution (solid line). The default parameter value S₁ = is indicated in both panels by a dashed vertical line.
在 SCM 模式下（D = 0），临界强迫值对 S ₁ 的依赖性。(a) F _w （红色方块）和 F _c （蓝色方块）与 S ₁ 的数值结果，以及 F _c 的精确解析解（蓝线）。当忽略冰厚度演变时，F _w 的精确解析解也包含在内（红线）。(b) 数值结果（方块）中滞后宽度ΔF 与 S ₁ 的关系，以及忽略冰厚度演变的解析解（实线）。默认参数值 S ₁ = 在两个面板中用虚线垂直线表示。

Citation: Journal of Climate 28, 10; 10.1175/JCLI-D-14-00654.1
引用：气候学杂志 28, 10; 10.1175/JCLI-D-14-00654.1

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The analogous relationship between F_w and S₁ is nonlinear, since F_w represents the transition from the ice-covered regime, in which the equations are nonlinear. However, we can consider the model as developed in section 2a, neglecting the nonlinear ice thickness evolution added in section 2b. In this case, only the surface albedo changes when T drops below T_m, and F_w (red line in Fig. 9a) takes a form similar to F_c. This leads to a hysteresis width (black line in Fig. 9b) of
F _w 和 S ₁ 之间的类比关系是非线性的，因为 F _w 代表了从冰覆盖状态的过渡，在该状态下方程是非线性的。然而，我们可以考虑在第 2a 节中开发的模型，忽略在第 2b 节中添加的非线性冰厚度演变。在这种情况下，当 T 下降到 T _m 以下时，只有表面反照率发生变化，F _w （图 9a 中的红线）呈现出类似于 F _c 的形式。这导致了滞后宽度（图 9b 中的黑线）为

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We therefore find that even in the simple case with no ice thickness evolution, increasing S₁ reduces the presence of instability. The derivation of Eqs. (13) and (14) is given in appendix B.
因此，我们发现即使在没有冰厚度演变的简单情况下，增加 S ₁ 也会减少不稳定性的存在。方程 (13) 和 (14) 的推导见附录 B。

Including the ice thickness evolution enhances this stabilizing effect. In this case F_w, and hence ΔF, is more sensitive overall to S₁ (red squares in Fig. 9a and black squares in Figs. 9b). Note that the diminished slope when S₁ > 220 W m⁻² in Figs. 9a and 9b is associated with the onset of seasonally ice-free conditions, analogous to Fig. 9d of Eisenman (2012). Since there is no smoothing of the albedo transition in the present model, further bifurcations associated with the transitions to and from seasonally ice-free conditions also occur (not shown), as in Fig. 9h of Eisenman (2012). There the presence of such bifurcations was suggested to be an artifact of a single-column representation. Consistent with this, they cease to occur when D > 0.05 W m⁻² K⁻¹ (not shown). However, since F_c and F_w are defined in terms of the presence of ice in winter, seasonally ice-free conditions do not influence the analysis here.
包括冰厚度演变增强了这种稳定效应。在这种情况下，F _w ，因此ΔF，对 S ₁ （图 9a 中的红色方块和图 9b 中的黑色方块）整体上更为敏感。注意，当 S ₁ > 220 W m ⁻² 时，图 9a 和 9b 中的斜率减小与季节性无冰条件的出现有关，类似于 Eisenman（2012）图 9d。由于当前模型中没有对反照率过渡进行平滑，因此与季节性无冰条件的过渡相关的进一步分岔也会发生（未显示），如 Eisenman（2012）图 9h 所示。那里建议这种分岔的存在是单列表示法的伪影。与此一致的是，当 D > 0.05 W m ⁻² K ⁻¹ 时，它们停止发生（未显示）。然而，由于 F _c 和 F _w 是根据冬季冰的存在来定义的，季节性无冰条件并不影响此处的分析。

Equation (13) indicates that F_c increases linearly with S₁ (Fig. 9a). This can be understood by considering that F = F_c is defined to be the point during a cooling scenario when ice first appears at the pole. This occurs when the winter minimum surface temperature crosses the freezing point (i.e., when ice appears on the coldest day of the year), at which point the ice–albedo feedback causes an abrupt transition to a perennially ice-covered state. In the perennially ice-free regime, the annual-mean value of E is linearly related to F and the seasonal amplitude of E is linearly related to S₁ because the governing equations are linear. The larger the seasonal amplitude of E, the warmer the annual-mean value of E can be and still have the winter minimum satisfy E < 0. Hence, larger values of S₁ are associated with larger values of F_c. This point is illustrated schematically in Fig. 10 (blue shading).
方程（13）表明 F _c 随着 S ₁ 线性增加（图 9a）。可以理解为 F = F _c 被定义为在冷却情景中冰首次出现在极地的时刻。这发生在冬季最低表面温度穿越冰点时（即在一年中最冷的一天出现冰的时候），此时冰-反照率反馈导致状态突然转变为永久冰覆盖状态。在永久无冰的状态下，E 的年均值与 F 线性相关，E 的季节振幅与 S ₁ 线性相关，因为控制方程是线性的。E 的季节振幅越大，E 的年均值可以越暖，仍然满足冬季最低值 E < 0。因此，较大的 S ₁ 值与较大的 F _c 值相关联。这个观点在图 10 中以示意图的形式展示（蓝色阴影）。

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Schematic illustrating the dependence of ΔF on S₁ in the SCM regime (D = 0). Turning on the seasonal cycle causes F_c (critical value of F in a cooling scenario; blue) to increase and F_w (critical value of F in a warming scenario; red) to decrease. The solid blue and red lines represent the enthalpy at the pole when S₁ = 0 and the colored shading represents the seasonal range at the pole when S₁ > 0. Ice is present when E < 0. The decrease in hysteresis width associated with the inclusion of seasonal variations is also indicated (from dark gray ΔF to light gray ΔF).
示意图说明了在 SCM 状态下（D = 0）ΔF 对 S ₁ 的依赖关系。季节循环的开启导致 F _c （冷却情景下的 F 临界值；蓝色）增加，而 F _w （加热情景下的 F 临界值；红色）减少。实线蓝色和红色表示当 S ₁ = 0 时极地的焓，而有色阴影表示当 S ₁ > 0 时极地的季节范围。当 E < 0 时，冰存在。与季节变化的纳入相关的滞后宽度的减少也被指示（从深灰色ΔF 到浅灰色ΔF）。

Citation: Journal of Climate 28, 10; 10.1175/JCLI-D-14-00654.1
引用：气候学杂志 28, 10; 10.1175/JCLI-D-14-00654.1

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An analogous argument applies to F_w (red shading in Fig. 10), which indicates the transition point when an initially ice-covered pole is subjected to increasing F. When the seasonal amplitude (S₁) increases, the warmest day of summer becomes ice free at an earlier point, that is, at a smaller value of F (red shading in Fig. 10). This argument for F_w applies exactly when ice thickness evolution is neglected. When it is included, the relationship becomes more complicated but the monotonic dependence remains (red squares in Fig. 9a).
类似的论点适用于 F _w （图 10 中的红色阴影），它表示当一个最初被冰覆盖的极点受到增加的 F 时的过渡点。当季节振幅 (S ₁ ) 增加时，夏季最温暖的一天在更早的时刻变得无冰，即在更小的 F 值时（图 10 中的红色阴影）。当忽略冰厚度演变时，F _w 的论点完全适用。当考虑冰厚度时，关系变得更加复杂，但单调依赖性仍然存在（图 9a 中的红色方块）。

Thinking in terms of the potential as in Fig. 8, one can visualize that a larger seasonal amplitude in forcing causes the system to climb higher onto either side of the well around a stable equilibrium. If the seasonal amplitude is sufficiently large, then the system will cross back and forth over the hump that separates the two stable equilibria during the course of each year, thereby having only a single seasonally varying equilibrium solution. A caveat with this explanation, however, is that the potential well associated with the time evolution of the system actually varies seasonally when S₁ > 0 and cannot strictly be visualized as remaining constant throughout the year.
从图 8 中考虑潜力的角度，可以想象更大的季节性强迫幅度使系统在稳定平衡的井的两侧爬升得更高。如果季节性幅度足够大，那么系统将在每年的过程中跨越分隔两个稳定平衡的隆起，从而仅具有一个季节性变化的平衡解。然而，这种解释的一个警告是，当 S ₁ > 0 时，与系统时间演化相关的势阱实际上是季节性变化的，不能严格地被视为在全年保持不变。

Another feature of the climate system that is represented in comprehensive GCMs but not in idealized models is high-frequency weather variability, which could be viewed as noise in the context of the model presented here. Although not included in this analysis (cf. Moon and Wettlaufer 2013), it is likely that the addition of a simple representation of noise would stabilize the model in a manner similar to increasing S₁, further diminishing the presence of bistability.
气候系统的另一个特征是在综合气候模型（GCMs）中体现但在理想化模型中未体现的高频天气变异性，这可以在此模型的背景下视为噪声。尽管在本分析中未包含（参见 Moon 和 Wettlaufer 2013），但很可能添加一个简单的噪声表示会以类似于增加 S ₁ 的方式稳定模型，进一步减少双稳态的存在。

e. Constructing full ΔF(D, S₁) space from limiting cases
e. 从极限情况构建完整的 ΔF(D, S ₁ ) 空间

We can consider the influence of both D > 0 and S₁ > 0 together by visualizing the combined influence of D and S₁ on the potential, adopting the crude approximation that the potential remains constant throughout the year when S₁ > 0, as discussed above. In this picture, larger values of S₁ will remove bistability in the presence of a larger hump in the potential well. This indicates that the influence of increasing D (decreasing the height of the hump) and increasing S₁ (increasing how large a hump the system can cross seasonally) can be added together to estimate ΔF. Hence we simply add the dependence on D shown in Fig. 7b to the dependence on S₁ shown in Fig. 9b to create a conceptually more tractable estimate of the main result of Fig. 6.
我们可以通过可视化 D > 0 和 S ₁ > 0 的综合影响来考虑它们的共同作用，采用粗略的近似，即当 S ₁ > 0 时，势能在全年保持不变，如上所述。在这种情况下，较大的 S ₁ 值将在势阱中存在更大隆起的情况下消除双稳态。这表明，增加 D（降低隆起的高度）和增加 S ₁ （增加系统季节性跨越隆起的能力）的影响可以相加来估计 ΔF。因此，我们简单地将图 7b 中对 D 的依赖性与图 9b 中对 S ₁ 的依赖性相加，以创建一个在概念上更易处理的图 6 主要结果的估计。

In other words, using the functional notation ΔF(D, S₁) as a shorthand, we approximate that
换句话说，使用函数符号 ΔF(D, S ₁ ) 作为简写，我们近似地认为

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Negative values of ΔF₀ are treated as zero, because they imply that S₁ is more than large enough for the system to cross the potential hump seasonally and hence that no bistability occurs. The first term on the right-hand side of Eq. (15) is given by the analytical solution from Eq. (10), the second term is the numerical result because no analytical solution of the nonlinear ice thickness evolution was obtained for F_w with D = 0, and the third term is the analytical expression Eq. (12). The result is shown in Fig. 11.
ΔF ₀ 的负值被视为零，因为它们意味着 S ₁ 对于系统跨越潜在的峰值是足够大的，因此不会发生双稳态。方程 (15) 右侧的第一项由方程 (10) 的解析解给出，第二项是数值结果，因为对于 D = 0 的 F _w 没有获得非线性冰厚演化的解析解，第三项是解析表达式方程 (12)。结果如图 11 所示。

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Approximation to ΔF(D, S₁) generated by adding the results from Figs. 7b and 9b.
通过将图 7b 和图 9b 的结果相加，生成对ΔF(D, S ₁ )的近似。

Citation: Journal of Climate 28, 10; 10.1175/JCLI-D-14-00654.1
引用：气候学杂志 28, 10; 10.1175/JCLI-D-14-00654.1

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It is not surprising that Fig. 11 does not exactly match Fig. 6, giving the approximation that was made by adding the two limiting cases, but the resemblance between the two suggests that the dependence of ΔF on D and S₁ can be qualitatively explained by considering the EBM and SCM limiting cases in isolation. Note that this approximation to ΔF (Fig. 11) is smaller than the full numerical solution for ΔF (Fig. 6) at all (D, S₁) points, which suggests that the two effects reinforce each other in reducing the bistability.
图 11 与图 6 不完全匹配并不令人惊讶，因为这是通过将两个极限情况相加而得出的近似值，但两者之间的相似性表明，ΔF 对 D 和 S ₁ 的依赖性可以通过单独考虑 EBM 和 SCM 的极限情况进行定性解释。请注意，这个对ΔF 的近似（图 11）在所有（D, S ₁ ）点上都小于ΔF 的完整数值解（图 6），这表明这两种效应在减少双稳态方面相互增强。