The albedo loss from the melting of the Greenland ice sheet and the social cost of carbon

Abstract

I extend the reduced Greenland ice sheet (GIS) model-module of DICE-GIS (Nordhaus, Proc Natl Acad Sci 25(116):12261–12269, 2019) by integrating snow-albedo feedback (SAF) and potential tipping of the ice sheet into the resuming DICE-GIS SAF model. The increasing global temperature no longer only results in the melting of the GIS, but also in albedo loss, which in turn impacts the strength of the SAF. As a consequence, global warming and the melting of the ice accelerate. The social cost of carbon (SCC) increases because the economic damages are not only related to intensified sea level rise, but also to accelerated global temperature rise. Accounting for the SAF raises the SCC from 37 to 41 $/t CO₂ in 2020, an increase of 11%. The temperature increase is the key channel through which the SAF impacts the SCC. The long-term volume of the GIS decreases by 2%, while the additional inclusion of tipping reduces it further by up to 35%.

Appendices

Appendix 1: Derivation of Eq. (8)

Equation: :

$$ A(t) = \left\{\begin{array}{ll} A_{0} & \text{for} ~ V(t) \ge ~ \delta_{V} \\ A_{0} ~ \frac{V(t)}{\eta ~V(t) + (1-\eta)~\frac{\delta_{V}}{100} ~ V_{0}} & \text{for} ~ V(t) < ~\delta_{V}. \end{array}\right. $$

Assumptions: :

• 1. $A(t) = \frac {V(t)}{H(t)}$

• 2. $\dot {H}(t)= \eta _{1} ~ \dot {V}(t)$ for all t with V (t) ≥ δ_V, whereby η₁ = 1/A₀

• 3. $\dot {H}(t)=\eta _{2} ~ \dot {V}(t)$ for all t with V (t) < δ_V, whereby η₂ = η/A₀ and η ∈ (0, 1)

Case: :

V (t) ≥ δ_V

I assume that A(t) = A₀, i.e., the surface area is the initial surface area and is constant over time. According to Assumption 2, the growth rate of H(t), denoted $\dot H(t)$, is given as

$$ \dot H(t) = \frac{\dot V(t)}{A_{0}}, $$

with $\dot V(t)$ denoting the growth rate of V (t). Integrating $\dot H(t)$ gives

$$ H(t) = \frac{V(t)}{A_{0}} + \sigma_{1}. $$

σ₁ is chosen such that

$$ H_{0} = \frac{V_{0}}{A_{0}}. $$

Hence, σ₁ = 0, and thus

$$ H(t) = \frac{V(t)}{A_{0}}. $$

Under Assumption 1, it follows for V (t) ≥ δ_V

$$ A(t) = \frac{V(t)}{H(t)} = \frac{V(t)}{\frac{1}{A_{0}} V(t)} = A_{0}. $$

Case: :

V (t) < δ_V

According to Assumption 3, it is

$$ \dot H(t) = \frac{\eta ~\dot V(t)}{A_{0}}. $$

Integrating $\dot H(t)$ gives

$$ H(t) = \frac{\eta ~ V(t)}{A_{0}} + \sigma_{2}. $$

σ₂ is chosen such that H(t) is a continuous function with t^∗ denoting the period when δ_V is reached. Hence, V (t^∗) = δ_V. This gives

$$ H(t^{*}) = \frac{V(t^{*})}{A_{0}} = \frac{\delta_{V}}{100} ~\frac{V_{0}}{A_{0}}. $$

Therefore, it follows for σ₂

$$ \sigma_{2} = \frac{\delta_{V}}{100} ~ \frac{V_{0}}{A_{0}} - \frac{\eta ~\frac{\delta_{V}}{100}~ V_{0}}{A_{0}} = \frac{(1-\eta)}{A_{0}}~ \frac{\delta_{V}}{100} V_{0}. $$

For t ≥ t^∗ it follows

$$ H(t) = \frac{\eta ~ V(t)}{A_{0}}+ \frac{(1-\eta)}{A_{0}} ~ \frac{\delta_{V}}{100} V_{0} . $$

Under Assumption 1, it follows for V (t) < δ_V

$$ A(t) = \frac{V(t)}{H(t)} = \frac{V(t)}{\frac{\eta}{A_{0}}V(t)+ \frac{(1-\eta)}{A_{0}} \frac{\delta_{V}}{100} V_{0} } = A_{0} ~ \frac{V(t)}{\eta ~V(t) + (1-\eta)~\frac{\delta_{V}}{100} ~ V_{0}} . $$

Appendix 2: Calibration tables

Table 2 Parameters volume and surface

Table 3 Parameters SAF

Table 4 Parameters tipping

Appendix 3: Non-optimal policy results

The results for the non-optimal policy are presented in the following. The main difference to the results for the optimal policy is mentioned.

3.1 A3.1 SCC under non-optimal policy

Figure 5a shows the SCC until 2100 and Fig. 5b presents the peak values for the SCC from 2350 to 2500. The gray graph refers to DICE-GIS, the orange graph to DICE-GIS SAF, and the green graph to DICE-GIS SAF with multiple tipping.

Fig. 5

SCC. a SCC until 2100. b Peak SCC

The peak SCC for DICE-GIS is remarkably lower than under the optimal policy. When accounting for the SAF, the SCC is a little higher and the difference in the SCC between DICE-GIS SAF and DICE-GIS SAF with multiple tipping is larger.

3.2 A3.2 GIS dynamics

Figure 6 displays the volume of the GIS on the left axis (solid lines) and the surface fraction of the GIS on the right axis (dashed lines) from 2015 to 3500. Compared to the optimal policy, the volume decreases faster and is significantly lower in the long run. Consequently, the surface fraction starts to decrease earlier. Under the non-optimal policy, multiple tipping is reached 100 years in advance in 2280. Moreover, the shrinking of the surface area for DICE-GIS SAF with multiple tipping begins 180 years earlier in 2380. For DICE-GIS SAF, the surface fraction begins to decline in 2480.

Fig. 6

Volume and surface fraction

Figure 7 presents the melt dynamics. In addition to the earlier passing of the tipping thresholds, the melt rate has a higher (more negative) magnitude, which leads to the greater volume loss (see Fig. 6). In contrast to the optimal policy, DICE-GIS SAF shows, in the long run, a slightly higher magnitude of the melt rate compared to DICE-GIS.

Fig. 7

Melt rate

3.3 A3.3 Temperature dynamics

Figure 8 shows the temperature increase T_globe on the left axis (solid lines) and the feedback parameter on the right axis (dashed lines). The peak increase in temperature is notably higher than under the optimal policy. When accounting for the SAF, the temperature increase shows only a very limited cooling after peaking. Hence, the temperature for DICE-GIS SAF remains at a relatively high level. The threshold κ_T is passed in 2085, 10 years earlier compared to the optimal policy. Surprisingly, the temperature increase is much higher for DICE-GIS SAF than for DICE-GIS SAF with multiple tipping.

Fig. 8

Feedback parameter

In general, the feedback parameter λ_FB decreases more than with the optimal policy. After the surface of the GIS starts to decrease in 2380 for DICE-GIS SAF with multiple tipping and in 2480 for DICE-GIS SAF (see also Fig. 6), λ_FB continues to decline steadily. In contrast to the optimal policy, there is no subsequent increase in the feedback parameter.

Appendix 4: Sensitivity analysis

The following appendix presents a comprehensive sensitivity analysis for the parameters mentioned in Section 4. The calculations are based on DICE-GIS SAF. Several calculations include tipping. Volume tipping refers to the solely passing of κ_V. While temperature tipping refers to the solely passing of κ_T. Multiple tipping refers to the passing of both thresholds κ_V and κ_T.

4.1 A4.1 Volume and surface: η

The following analysis is based on DICE-GIS SAF with multiple tipping, because for DICE-GIS SAF, the volume does not fall below δ_V = 80. Figure 9 shows the SCC in dependence to η. Varying η in the range of 0.25 to 0.75 gives no significant change in the SCC in the year 2100. There is an absolute deviation in the peak SCC of − 28.20 to 18.55 $/t CO₂. This gives a relative deviation of − 0.65 to 0.40%.

Fig. 9

Multiple tipping—SCC depending on η; deviation reference η = 0.45

Figure 10 displays the volume in dependence to η. The volume in 3500 gives an absolute deviation from − 0.94 to 1.40%p (percentage points).

Fig. 10

Multiple tipping—volume depending on η; deviation reference η = 0.45

The higher η, the more convex is the relation between the loss of the volume and the decrease of the surface fraction. Hence, the higher η, the higher is the surface fraction for a given volume. This implies less total albedo loss and thus a lower decrease in the feedback parameter after passing δ_V. Therefore, the temperature increase is negatively correlated to η. Hence, the SCC is relatively lower while the volume is relatively higher.

I conclude that the SCC and the volume are relatively robust to the variation of η.

4.2 A4.2 Volume and surface: δ _V

The following analysis is based on DICE-GIS SAF with volume tipping. There is no significant deviation in the SCC when varying δ_V from 76 to 85. The absolute deviation of the peak SCC is between − 0.2 to 0.7 $/tCO₂.

Figure 11 shows the impact on the volume. The absolute deviation in 3500 is in the range of − 0.26 to 0.09%p.

Fig. 11

Volume tipping—volume depending on δ_V; deviation reference δ_V = 80

Note, that the volume starts to deviate earlier for higher values of δ_V, because as the surface fraction declines, there is total albedo loss and the feedback parameter decreases again (see Fig. 4b). This, in turn, accelerates the temperature increase and promotes the melting of the GIS.

In addition, I vary δ_V from 99.9 to 91 to investigate the impact on DICE-GIS SAF as well. For δ_V = 99.9, the surface fraction starts to decrease almost immediately as the volume melts.

Figure 12 presents the SCC depending on δ_V. The SCC in 2100 shows an absolute deviation of 2.29 $/t CO₂, or relatively 0.7%. The peak SCC deviates up to 131.63 $/t CO₂, or 3.1%.

Fig. 12

DICE-GIS SAF—SCC depending on δ_V; deviation reference δ_V = 80

Figure 13 shows the volume in dependence to δ_V. The GIS starts to melt comparatively earlier for higher values of δ_V, which reveals a similar pattern as in Fig. 11. The volume in 3500 decreases by up to − 0.83%p.

Fig. 13

DICE-GIS SAF—volume depending on δ_V; deviation reference δ_V = 80

Consequently, the volume is relatively robust to the variation of δ_V. This holds for DICE-GIS SAF as well as for DICE-GIS SAF with volume tipping. The SCC is sensitive for DICE-GIS SAF under δ_V ≥ 91, but not for volume tipping under 75 ≤ δ_V ≤ 85.

4.3 A4.3 SAF: λ _SAF

The following analysis is based on DICE-GIS SAF. Figure 14 presents the SCC in dependence to λ_SAF in the range of 0.25 to 0.75 (Flanner et al. 2011; Duan et al. 2019; Qu and Hall 2014; Winton 2006). The SCC in 2100 differs between − 23.83 and 27.47 $/tCO₂, or − 7.5 to 8.5%. The deviation of the peak SCC is − 359.90 to 431.51 $/tCO₂, or − 8.5 to 10.2%.

Fig. 14

DICE-GIS SAF—SCC depending on λ_SAF; deviation reference λ_SAF = 0.5

Figure 15 shows the influence of the absolute contribution of the SAF to the climate feedback parameter λ_SAF. The higher λ_SAF, the stronger is the decreasing impact on the global feedback parameter λ_FB. Thereby, the course remains almost robust. The feedback parameter in 3500 is in the range of 1.06 to 1.15, a relative deviation of − 4.6 to 3.6%.

Fig. 15

DICE-GIS SAF—λ_FB depending on λ_SAF; deviation reference λ_SAF = 0.5

I deduce that the SCC and the feedback parameter are sensitive to λ_SAF. This underlines the fact that the SAF plays an important role for the climate and has a significant impact on the SCC.

4.4 A4.4 SAF: γ

The following analysis is based on DICE-GIS SAF. Figure 16 displays the SCC in dependence to γ. The SCC in 2100 varies between − 5.89 and 6.09 $/tCO₂, or − 1.8 to 1.9%. The peak SCC deviates from − 89.77 to 93.77 $/tCO₂, or − 2.1 to 2.2%.

Fig. 16

DICE-GIS SAF—SCC depending on γ; deviation reference γ = − 80,000

I conclude that the SCC is sensitive to γ, which underlines the impact of the SAF.

4.5 A4.5 Tipping: κ _V

The following analysis is based on DICE-GIS SAF with volume tipping. Figure 17 visualizes the SCC with κ_V varying from 95 to 85. There is no significant impact on the SCC in 2100. The peak SCC deviates in the range of − 2.96 to 22.65 $/tCO₂, or − 0.06 to 0.5%. The volume deviates in the range of − 0.05 to 0.06%p.

Fig. 17

Volume tipping—SCC depending on κ_V deviation reference κ_V = 90

I derive a limited sensitivity of the SCC to κ_V, while the volume is robust. This is because κ_V is reached very late (see Fig. 2a).

4.6 A4.6 Tipping: κ _T

The following analysis is based on DICE-GIS SAF with temperature tipping. Figure 18 presents the SCC with κ_T varying from 1.5 to 4.5 °C (IPCC 2019; Robinson et al. 2012; Ridley et al. 2009). The SCC in 2100 falls by up to − 1.52 $/t CO₂, or 0.04%. The peak SCC deviates in the range of − 20.64 to 0.50 $/tCO₂, or − 0.5 to 0.01%.

Fig. 18

Temperature tipping—SCC depending on κ_T; deviation reference κ_T = 3.4

Figure 19 shows the volume depending on κ_T and depicts a similar pattern. The volume is only sensitive for κ_T ≥ 3.6 °C and deviates up to 13.0%p.

Fig. 19

Temperature tipping—volume depending on κ_T; deviation reference κ_T = 3.4

Since the thresholds for κ_T < 3.6 °C are all reached very early (see Fig. 4b), temperature tipping starts at about the same time for 1.5 ≤ κ_T < 3.6. Therefore, neither the SCC nor the volume shows a significant deviation. For κ_T ≥ 3.6 °C, the SCC and the volume are sensitive as the timing of passing the threshold varies.

4.7 A4.7 Tipping: ρ

The following analysis is based on DICE-GIS SAF with multiple tipping. Figure 20 shows the SCC in dependence to ρ varying from 1.5 to 2.5. There is no significant variation in the SCC in 2100. The peak SCC deviates in the range of − 50.23 to 89.60 $/tCO₂, or by − 1.2 to 2.1%.

Fig. 20

Multiple tipping—SCC depending on ρ; deviation reference ρ = 2

The volume deviates in the range of − 0.19 to 0.18%p. I deduce that the SCC and particularly the volume are rather robust to ρ.

4.8 A4.8 Tipping: ψ

The following analysis incorporates DICE-GIS SAF with volume, temperature and multiple tipping. ψ is varied in the range of 1.5 to 4. Figure 21 presents the SCC in variation to ψ.

Fig. 21

SCC depending on ψ; deviation reference ψ = 2. a Volume tipping. b Temperature tipping. c Multiple tipping. d Multiple tipping

For DICE-GIS SAF with volume tipping, there is no deviation in 2100. But, the peak SCC varies in the range of − 1.37 to 6.66 $/tCO₂, or − 0.03 to 0.16%. Surprisingly the SCC for ψ = 1.75 initially shows a positive deviation followed by a negative deviation. For DICE-GIS SAF with temperature tipping, the SCC in 2100 varies from − 0.75 to 2.95 $/tCO₂, or − 0.02 to 0.09%. The deviation of the peak SCC is between − 10.63 to 171.29 $/tCO₂, or − 0.2 to 4.0%. For DICE-GIS SAF with multiple tipping and ψ ≤ 3, the SCC in 2100 varies from − 0.79 to 2.27 $/tCO₂, or − 0.2 to 0.7%. The deviation of the peak SCC is − 73.94 to 467.10 $/tCO₂, or − 1.7 to 10.8%. The SCC is highly sensitive for ψ > 3, exploding up to 3.9 million $/tCO₂.

Figure 22 shows the volume in variation to ψ. Accounting for volume tipping, the volume in 3500 is in the range of 62.16 to 77.54, or − 11.92 to 3.46%p. Accounting for temperature tipping, it is in the range of 48.99 to 74.39, or − 19.27 to 6.12%p. Accounting for multiple tipping, it is in the range of 19.06 to 68.13, or − 34.02 to 15.05%p.

Fig. 22

Volume depending on ψ; deviation reference ψ = 2. a Volume tipping. b Temperature tipping. c Multiple tipping

I conclude that, surprisingly, there is only a very small impact for volume tipping. Temperature tipping is quite sensitive to ψ. While accounting for multiple tipping is the most sensitive, because ψ is exponentiated with ρ (see Eq. (15)).