Sea-level rise
The sea-level-rise (SLR) component represents thermal expansion, glaciers, the Greenland ice sheet (GIS), the Antarctic ice sheet (AIS), and future changes in land-water storage (LWS). The formulation is deliberately small and deterministic so it can be evaluated both by the MIMOSA simulator and as part of the Pyomo optimisation model.
All component values are expressed as metres above the 1900 global mean sea-level reference. The values at the model start are linearly interpolated between zero in 1900 and the central 2025 estimates below. This makes the reference year explicit and prevents a run starting before 2025 from using 2025 sea-level values.
The component option projection selects a coherent low, central, or
high deterministic response set. The sets use identical historical
initial conditions, but different sensitivities, response times, and AIS
fast-response assumptions. The default is central.
The projection can be selected before model construction with:
Reference years and initial values
These values are shared by all three projections. Initial sea-level contributions are relative to the reference year and specified at the historical calibration year. At other start years, the model scales the initial values as described above. The land-water-storage anomaly starts at zero.
| Symbol | Meaning | Units | Value |
|---|---|---|---|
| \(y_{\mathrm{ref}}\) | Sea-level reference year | year | 1900 |
| \(y_{\mathrm{init}}\) | Historical calibration year | year | 2025 |
| \(\text{SLR}_{\mathrm{thermal,fast,init}}\) | Initial fast thermal-expansion contribution | m | 0.045 |
| \(\text{SLR}_{\mathrm{thermal,slow,init}}\) | Initial slow thermal-expansion contribution | m | 0.025 |
| \(\text{SLR}_{\mathrm{GL,init}}\) | Initial glacier contribution | m | 0.09 |
| \(\text{SLR}_{\mathrm{GIS,init}}\) | Initial Greenland contribution | m | 0.045 |
| \(\text{SLR}_{\mathrm{AIS,init}}\) | Initial Antarctic contribution | m | 0.025 |
| \(T_{\mathrm{A,init}}\) | Initial Antarctic ocean-temperature proxy | °C above pre-industrial | 0.3 |
In the equations below, \(t\) indexes model time steps, \(\Delta t\) is the elapsed time in years, and \(\href{./#var__temperature}{T_{t-1}}\) is global mean warming above pre-industrial at the preceding step. A starred SLR value is the long-term equilibrium contribution at a fixed temperature; the actual contribution can lag behind that equilibrium.
Thermal expansion
Thermal expansion is represented by a fast and a slow response box. Each box relaxes towards a temperature-dependent equilibrium:
\[
\begin{aligned}
\href{./#var__slr_thermal_fast}{\text{SLR}_{\mathrm{thermal,fast},t}}
&=\href{./#slrcoef__slr_thermal_fast_sensitivity}{\beta_{\mathrm{fast}}} \href{./#var__temperature}{T_{t-1}} \\
&\quad+\left(\href{./#var__slr_thermal_fast}{\text{SLR}_{\mathrm{thermal,fast},t-1}}-
\href{./#slrcoef__slr_thermal_fast_sensitivity}{\beta_{\mathrm{fast}}} \href{./#var__temperature}{T_{t-1}}\right)
e^{-\Delta t/\href{./#slrcoef__slr_thermal_fast_timescale}{\tau_{\mathrm{fast}}}}.
\end{aligned}
\]
\[
\begin{aligned}
\href{./#var__slr_thermal_slow}{\text{SLR}_{\mathrm{thermal,slow},t}}
&=\href{./#slrcoef__slr_thermal_slow_sensitivity}{\beta_{\mathrm{slow}}} \href{./#var__temperature}{T_{t-1}} \\
&\quad+\left(\href{./#var__slr_thermal_slow}{\text{SLR}_{\mathrm{thermal,slow},t-1}}-
\href{./#slrcoef__slr_thermal_slow_sensitivity}{\beta_{\mathrm{slow}}} \href{./#var__temperature}{T_{t-1}}\right)
e^{-\Delta t/\href{./#slrcoef__slr_thermal_slow_timescale}{\tau_{\mathrm{slow}}}}.
\end{aligned}
\]
The total thermal-expansion contribution is the sum of the fast and slow contributions:
\[
\href{./#var__slr_thermal}{\text{SLR}_{\mathrm{thermal},t}}=
\href{./#var__slr_thermal_fast}{\text{SLR}_{\mathrm{thermal,fast},t}}+
\href{./#var__slr_thermal_slow}{\text{SLR}_{\mathrm{thermal,slow},t}}.
\]
The exact exponential update makes the response independent of the chosen numerical time step for a constant temperature forcing.
This two-timescale formulation is a reduced-form approximation chosen for MIMOSA, rather than a direct reproduction of an ocean model. The first-order relaxation form follows BRICK v0.2 (Wong et al., 2017, Sect. 3.2.4, Eqs. 9--10). Splitting it into fast and slow boxes represents the upper- and deep-ocean distinction in SURFER v2.0 (Martínez Montero et al., 2022, Sects. 2.3.1 and 2.4.3). MIMOSA's equilibrium sensitivities and response times are calibrated jointly against the AR6 thermal-expansion contributions in Table 9.9 of Fox-Kemper et al. (2021); they are not the coefficients from either BRICK or SURFER.
Coefficients
| Symbol | Meaning | Units | Low | Central (default) | High |
|---|---|---|---|---|---|
| \(\beta_{\mathrm{fast}}\) | Fast equilibrium sensitivity | m/°C | 0.075 | 0.08 | 0.1 |
| \(\beta_{\mathrm{slow}}\) | Slow equilibrium sensitivity | m/°C | 0.255 | 0.27 | 0.34 |
| \(\tau_{\mathrm{fast}}\) | Fast response time | years | 42.5 | 40 | 30 |
| \(\tau_{\mathrm{slow}}\) | Slow response time | years | 425 | 400 | 300 |
Glaciers
The glacier contribution relaxes towards the capped equilibrium proposed by Martínez Montero et al. (2022):
\[
\text{SLR}^*_{\mathrm{GL}}(\href{./#var__temperature}{T_{t-1}})=\href{./#slrcoef__slr_gsic_total_ice}{P_{\mathrm{GL}}}
\tanh\left(\frac{\href{./#var__temperature}{T_{t-1}}}{\href{./#slrcoef__slr_gsic_temp_sensitivity}{\zeta_{\mathrm{GL}}}}\right),
\]
\[
\href{./#var__slr_cumgsic}{\text{SLR}_{\mathrm{GL},t}}=\text{SLR}^*_{\mathrm{GL}}(\href{./#var__temperature}{T_{t-1}})+
\left(\href{./#var__slr_cumgsic}{\text{SLR}_{\mathrm{GL},t-1}}-\text{SLR}^*_{\mathrm{GL}}(\href{./#var__temperature}{T_{t-1}})\right)
e^{-\Delta t/\href{./#slrcoef__slr_gsic_timescale}{\tau_{\mathrm{GL}}}}.
\]
This replaces the previous formulation, in which all glaciers eventually melted under any sustained temperature above -1 degree C.
Coefficients
| Symbol | Meaning | Units | Low | Central (default) | High |
|---|---|---|---|---|---|
| \(\zeta_{\mathrm{GL}}\) | Equilibrium temperature scale | °C | 2.4 | 2 | 1.6 |
| \(\tau_{\mathrm{GL}}\) | Glacier response time | years | 120 | 110 | 140 |
The following constants are shared by all three projections.
| Symbol | Meaning | Units | Value |
|---|---|---|---|
| \(P_{\mathrm{GL}}\) | Glacier sea-level potential | m | 0.32 |
Greenland ice sheet
The Greenland ice sheet contributes to sea-level rise as it loses ice. MIMOSA uses a SIMPLE-style delayed equilibrium: the temperature at the preceding step sets a long-term equilibrium contribution, and the actual contribution approaches that target gradually.
The equilibrium curve uses the logistic function
\[
\sigma(x)=\frac{1}{1+e^{-x}},
\]
a smooth transition from 0 to 1. Here \(x\) is a dimensionless argument. The equilibrium SLR contribution is normalised to zero at zero warming and approaches the ice sheet's sea-level potential at high warming:
\[
\text{SLR}^*_{\mathrm{GIS}}(\href{./#var__temperature}{T_{t-1}})=\href{./#slrcoef__slr_gis_total_ice}{P_{\mathrm{GIS}}}
\frac{\sigma((\href{./#var__temperature}{T_{t-1}}-\href{./#slrcoef__slr_gis_threshold}{T_c})/\href{./#slrcoef__slr_gis_transition_width}{w})-\sigma(-\href{./#slrcoef__slr_gis_threshold}{T_c}/\href{./#slrcoef__slr_gis_transition_width}{w})}
{1-\sigma(-\href{./#slrcoef__slr_gis_threshold}{T_c}/\href{./#slrcoef__slr_gis_transition_width}{w})},
\]
The response time \(\tau_{\mathrm{GIS}}\), measured in years, determines how quickly Greenland approaches this equilibrium. It decreases with warming, so higher temperatures lead to a faster response:
\[
\tau_{\mathrm{GIS}}(\href{./#var__temperature}{T_{t-1}})=\href{./#slrcoef__slr_gis_base_timescale}{\tau_0}e^{-\href{./#slrcoef__slr_gis_timescale_sensitivity}{\gamma} \href{./#var__temperature}{T_{t-1}}}.
\]
The actual Greenland SLR contribution is updated each time step as
\[
\begin{aligned}
\href{./#var__slr_cumgis}{\text{SLR}_{\mathrm{GIS},t}}
&= \text{SLR}^*_{\mathrm{GIS}}(\href{./#var__temperature}{T_{t-1}}) \\
&\quad + \left(\href{./#var__slr_cumgis}{\text{SLR}_{\mathrm{GIS},t-1}}
- \text{SLR}^*_{\mathrm{GIS}}(\href{./#var__temperature}{T_{t-1}})\right)
e^{-\Delta t/\tau_{\mathrm{GIS}}(\href{./#var__temperature}{T_{t-1}})}.
\end{aligned}
\]
Thus, only part of the gap to equilibrium closes within a model time step. For example, with a constant temperature and response time, about 63% of that gap closes after one response time. The actual SLR contribution, rather than the equilibrium target, is added to total sea-level rise.
Coefficients
| Symbol | Meaning | Units | Low | Central (default) | High |
|---|---|---|---|---|---|
| \(T_c\) | Equilibrium transition temperature | °C above pre-industrial | 2 | 1.8 | 1.4 |
| \(w\) | Equilibrium transition width | °C | 0.7 | 0.6 | 0.5 |
| \(\tau_0\) | Response time at zero warming | years | 6500 | 6000 | 4000 |
| \(\gamma\) | Response-time sensitivity to warming | 1/°C | 0.2 | 0.3 | 0.4 |
The following constants are shared by all three projections.
| Symbol | Meaning | Units | Value |
|---|---|---|---|
| \(P_{\mathrm{GIS}}\) | Greenland sea-level potential | m | 7.3 |
Antarctic ice sheet
A single lagged Antarctic subsurface-ocean temperature proxy drives the AIS response. A smooth threshold term represents the possibility of faster ice loss at high warming while retaining differentiability for optimisation.
The function \(\sigma\) is the same logistic function defined in the Greenland section above.
\[
\href{./#var__slr_antarctic_ocean_temp}{T_{A,t}}=\href{./#slrcoef__slr_ais_ocean_temp_scaling}{\lambda_A} \href{./#var__temperature}{T_{t-1}}+
\left(\href{./#var__slr_antarctic_ocean_temp}{T_{A,t-1}}-\href{./#slrcoef__slr_ais_ocean_temp_scaling}{\lambda_A} \href{./#var__temperature}{T_{t-1}}\right)e^{-\Delta t/\href{./#slrcoef__slr_ais_ocean_temp_timescale}{\tau_A}},
\]
\[
\href{./#var__slr_cumais}{\text{SLR}_{\mathrm{AIS},t}}=\href{./#var__slr_cumais}{\text{SLR}_{\mathrm{AIS},t-1}}+\Delta t
\left[\href{./#slrcoef__slr_ais_background_rate}{r_0}+\href{./#slrcoef__slr_ais_temp_sensitivity}{r_1}\href{./#var__slr_antarctic_ocean_temp}{T_{A,t}}+\href{./#slrcoef__slr_ais_fast_rate}{r_f}\sigma((\href{./#var__slr_antarctic_ocean_temp}{T_{A,t}}-\href{./#slrcoef__slr_ais_fast_threshold}{T_{crit}})/\href{./#slrcoef__slr_ais_fast_transition_width}{w_A})\right]
\left(1-\frac{\href{./#var__slr_cumais}{\text{SLR}_{\mathrm{AIS},t-1}}}{\href{./#slrcoef__slr_ais_total_ice}{P_{\mathrm{AIS}}}}\right).
\]
The scaling and delay between global surface warming and Antarctic subsurface-ocean warming are motivated by LARMIP-2 (Levermann et al., 2020, Sect. 2.2). The AIS rate equation itself is a deliberately simpler MIMOSA parameterisation, not a reproduction of LARMIP-2 or SURFER. Its smooth fast term is inspired by the threshold representation of uncertain rapid Antarctic disintegration in Wong et al. (2017). The ordinary background and temperature-response rates are calibrated against the AR6 AIS contributions in Table 9.9. The high parameter set is intended as a low-likelihood, high-impact sensitivity case and not as the upper endpoint of the AR6 likely range.
Coefficients
| Symbol | Meaning | Units | Low | Central (default) | High |
|---|---|---|---|---|---|
| \(\lambda_A\) | Ocean-temperature scaling | dimensionless | 0.45 | 0.6 | 0.8 |
| \(\tau_A\) | Ocean-temperature response time | years | 50 | 30 | 20 |
| \(r_0\) | Background ice-loss rate | m/year | 0.0009 | 0.0011 | 0.0012 |
| \(r_1\) | Ice-loss sensitivity to ocean temperature | m/(year · °C) | 0.0002 | 0.00025 | 0.00035 |
| \(r_f\) | Additional fast-response rate | m/year | 0 | 0.005 | 0.012 |
| \(T_{crit}\) | Fast-response threshold temperature | °C above pre-industrial | 3 | 2.5 | 1.8 |
| \(w_A\) | Fast-response transition width | °C | 0.2 | 0.15 | 0.12 |
The following constants are shared by all three projections.
| Symbol | Meaning | Units | Value |
|---|---|---|---|
| \(P_{\mathrm{AIS}}\) | Effective vulnerable Antarctic ice stock | m | 5 |
Land-water storage
AR6 projects an approximately scenario-independent land-water contribution of 0.03 m between 1995--2014 and 2100. Because the historical contribution is already implicit in MIMOSA's 2025 initial total, only its future anomaly is added explicitly:
\[
\href{./#var__slr_cumlws}{\text{SLR}_{\mathrm{LWS},t}}=\href{./#var__slr_cumlws}{\text{SLR}_{\mathrm{LWS},t-1}}+\Delta t\,\href{./#slrcoef__slr_lws_rate}{r_{\mathrm{LWS}}},
\qquad \href{./#slrcoef__slr_lws_rate}{r_{\mathrm{LWS}}}=0.0004\ \mathrm{m\,yr^{-1}}.
\]
Coefficients
| Symbol | Meaning | Units | Value |
|---|---|---|---|
| \(r_{\mathrm{LWS}}\) | Future land-water-storage contribution rate | m/year | 0.0004 |
Total sea-level rise
\[
\href{./#var__total_SLR}{\text{SLR}_t}=\href{./#var__slr_thermal}{\text{SLR}_{\mathrm{thermal},t}}+\href{./#var__slr_cumgsic}{\text{SLR}_{\mathrm{GL},t}}
+\href{./#var__slr_cumgis}{\text{SLR}_{\mathrm{GIS},t}}+\href{./#var__slr_cumais}{\text{SLR}_{\mathrm{AIS},t}}+\href{./#var__slr_cumlws}{\text{SLR}_{\mathrm{LWS},t}}.
\]
Calibration against IPCC AR6
The process papers motivate the compact equation forms; they do not imply that every coefficient is transferred unchanged. The three coherent parameter sets are outcome-calibrated against AR6 WGI Table 9.9 and the warming-level assessment in Chapter 12. Consequently, increasing the glacier or ordinary AIS response remains consistent with the cited equation structure while bringing total SLR into the assessed AR6 ranges. The following benchmark prescribes a linear warming path from 1.27 degrees C in 2025 to the stated 2100 warming level. Values are metres relative to 1900:
| 2100 warming | AR6 median [likely range] | low | central | high |
|---|---|---|---|---|
| 2 degrees C | 0.668 [0.558--0.848] | 0.547 | 0.619 | 0.825 |
| 3 degrees C | 0.778 [0.658--0.968] | 0.644 | 0.741 | 1.162 |
| 4 degrees C | 0.858 [0.738--1.068] | 0.737 | 0.858 | 1.531 |
The MIMOSA columns in this table and the component table below are generated directly from the current component equations. After changing the SLR formulation or parameters, regenerate both tables from the repository root:
A model test compares the committed CSV files with a fresh calculation so that outdated documentation is detected automatically.
AR6 Chapter 12 reports the warming-level values relative to 1995--2014. The table adds the assessed 0.158 m rise from 1900 to 1995--2014, following the AR6 Summary for Policymakers. The comparison is a calibration diagnostic, not a claim that a warming level uniquely determines SLR: the preceding temperature pathway and rate of warming also affect the lagged components.
Under this diagnostic, the low set tracks the lower edge of the AR6 likely range and central remains within it. High exceeds the likely range at 3 and 4 degrees C by design, representing rapid Antarctic loss. For transparency, the central 2100 component values at 2, 3, and 4 degrees C respectively are:
| Component | 2 degrees C | 3 degrees C | 4 degrees C |
|---|---|---|---|
| Thermal expansion | 0.217 | 0.277 | 0.338 |
| Glaciers | 0.151 | 0.167 | 0.177 |
| Greenland | 0.102 | 0.143 | 0.187 |
| Antarctica | 0.12 | 0.122 | 0.125 |
| Land-water storage | 0.03 | 0.03 | 0.03 |
References:
- Fox-Kemper et al. (2021), IPCC AR6 WGI Chapter 9.
- IPCC AR6 WGI Chapter 12, climate impact-driver projections.
- IPCC AR6 WGI Summary for Policymakers.
- Bakker, Applegate and Keller (2016), SIMPLE.
- Martínez Montero et al. (2022), SURFER v2.0.
- Wong et al. (2017), BRICK v0.2.
- Levermann et al. (2020), LARMIP-2.
- Wong, Bakker and Keller (2017), Antarctic fast dynamics.
Source code in mimosa/components/sealevelrise.py
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