The sea-level rise (SLR) module is based on the AD-RICE 2012 model (Kelly de Bruin, 2014),
which in itself is based on RICE 2010. It consists of sea-level rise contributions from thermal expansion, glaciers and small ice caps (GSIC),
and the Greenland ice sheet (GIS). Sea-level rise from Antarctica is not included due to the high uncertainty involved.
Thermal expansion
The sea-level rise due to thermal expansion is calculated as follows:
\[
\text{SLR}_{\text{thermal}, t} = \text{SLR}_{\text{thermal},t-1} \cdot (1 - a)^{\frac{\Delta t}{10}} + \text{temperature}_{t-1} \cdot a \cdot \tfrac{\Delta t}{10} \cdot \text{equil. rate}_{\text{thermal}}
\]
The initial sea-level rise due to thermal expansion (at time \(t=0\)) is calculated as:
\[
\text{SLR}_{\text{thermal},0} = \text{SLR}_{\text{thermal,init}} + a \cdot (\text{temperature}_{t=0} \cdot \text{equil. rate}_{\text{thermal}} - \text{SLR}_{\text{thermal,init}})
\]
with the following parameters:
| Name |
Description |
Value |
| \(a\) |
Adjustment rate for thermal expansion |
0.024076 |
| \(\text{equil. rate}_{\text{thermal}}\) |
Equilibrium rate for thermal expansion |
0.5 m / °C |
| \(\text{SLR}_{\text{thermal,init}}\) |
Initial sea-level rise due to thermal expansion |
0.092067 m |
Glaciers and small ice caps (GSIC)
Next, melting of glaciers and small ice caps (GSIC) contribute to sea-level rise according to:
\[
\begin{align*}
\text{SLR}_{\text{GSIC}, t} &= \text{ SLR}_{\text{GSIC}, t-1}\\
+\ &\text{melt rate}\cdot \Delta t \cdot \frac{\text{total ice}_{\text{GSIC}} - \text{SLR}_{\text{GSIC}, t-1}}{\text{total ice}_{\text{GSIC}}} \cdot (\text{temperature}_{t-1} - \text{equil. temp}_{\text{GSIC}})
\end{align*}
\]
with as initial value:
\[
\text{SLR}_{\text{GSIC},0} = 0.015
\]
and with the following parameters:
| Name |
Description |
Value |
| \(\text{melt rate}\) |
Melt rate |
0.0008 m / year |
| \(\text{total ice}_{\text{GSIC}}\) |
Total ice |
0.26 m |
| \(\text{equil. temp}_{\text{GSIC}}\) |
Equilibrium temperature |
-1 °C |
Greenland ice sheet (GIS)
The Greenland ice sheet (GIS) contributes to sea-level rise according to:
\[
\begin{align*}
\text{SLR}_{\text{GIS}, t} &= \text{SLR}_{\text{GIS}, t-1}\\
+\ &\tfrac{\Delta t}{10} \cdot \tfrac{1}{100} \cdot (\text{melt rate above thresh} \cdot \text{temperature}_{t-1} + \text{init melt rate}) \\
& \cdot \left(1 - \frac{\text{SLR}_{\text{GIS}, t-1}}{\text{init ice volume}}\right)
\end{align*}
\]
with as initial value:
\[
\text{SLR}_{\text{GIS},0} = 0.006
\]
and with the following parameters:
| Name |
Description |
Value |
| \(\text{melt rate above thresh}\) |
Melt rate above threshold |
1.1186 m / year |
| \(\text{init melt rate}\) |
Initial melt rate |
0.6 m / year |
| \(\text{init ice volume}\) |
Initial ice volume |
7.3 m |
Total sea-level rise
The total sea-level rise is the sum of the contributions from thermal expansion, GSIC, and GIS:
\[
\text{SLR}_t = \text{SLR}_{\text{thermal},t} + \text{SLR}_{\text{GSIC},t} + \text{SLR}_{\text{GIS},t}
\]
Source code in mimosa/components/sealevelrise.py
| def get_constraints(m: AbstractModel) -> Sequence[GeneralConstraint]:
"""
The sea-level rise (SLR) module is based on the AD-RICE 2012 model ([Kelly de Bruin, 2014](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2600006)),
which in itself is based on RICE 2010. It consists of sea-level rise contributions from thermal expansion, glaciers and small ice caps (GSIC),
and the Greenland ice sheet (GIS). Sea-level rise from Antarctica is not included due to the high uncertainty involved.
# Thermal expansion
:::mimosa.components.sealevelrise.slr_thermal_expansion
options:
show_source: false
The initial sea-level rise due to thermal expansion (at time $t=0$) is calculated as:
:::mimosa.components.sealevelrise.slr_thermal_expansion_init
options:
show_source: false
with the following parameters:
| Name | Description | Value |
| --- | --- | --- |
| $a$ | Adjustment rate for thermal expansion | 0.024076 |
| $\\text{equil. rate}_{\\text{thermal}}$ | Equilibrium rate for thermal expansion | 0.5 m / °C |
| $\\text{SLR}_{\\text{thermal,init}}$ | Initial sea-level rise due to thermal expansion | 0.092067 m |
# Glaciers and small ice caps (GSIC)
:::mimosa.components.sealevelrise.slr_gsic
options:
show_source: false
with as initial value:
$$
\\text{SLR}_{\\text{GSIC},0} = 0.015
$$
and with the following parameters:
| Name | Description | Value |
| --- | --- | --- |
| $\\text{melt rate}$ | Melt rate | 0.0008 m / year |
| $\\text{total ice}_{\\text{GSIC}}$ | Total ice | 0.26 m |
| $\\text{equil. temp}_{\\text{GSIC}}$ | Equilibrium temperature | -1 °C |
# Greenland ice sheet (GIS)
:::mimosa.components.sealevelrise.slr_gis
options:
show_source: false
with as initial value:
$$
\\text{SLR}_{\\text{GIS},0} = 0.006
$$
and with the following parameters:
| Name | Description | Value |
| --- | --- | --- |
| $\\text{melt rate above thresh}$ | Melt rate above threshold | 1.1186 m / year |
| $\\text{init melt rate}$ | Initial melt rate | 0.6 m / year |
| $\\text{init ice volume}$ | Initial ice volume | 7.3 m |
# Total sea-level rise
The total sea-level rise is the sum of the contributions from thermal expansion, GSIC, and GIS:
$$
\\text{SLR}_t = \\text{SLR}_{\\text{thermal},t} + \\text{SLR}_{\\text{GSIC},t} + \\text{SLR}_{\\text{GIS},t}
$$
"""
# Parameters and variables necessary for sea level rise
m.slr_thermal = Var(m.t, within=NonNegativeReals, units=quant.unit("m"))
m.slr_thermal_equil = Param(initialize=0.5) # Equilibrium
m.slr_thermal_init = Param(
initialize=0.0920666936642
) # Initial SLR due to thermal expansion
m.slr_thermal_adjust_rate = Param(initialize=0.024076141150722) # Adjustment rate
m.slr_cumgsic = Var(m.t, within=NonNegativeReals, units=quant.unit("m"))
m.slr_gsic_melt_rate = Param(initialize=0.0008) # Melt rate
m.slr_gsic_total_ice = Param(initialize=0.26) # Total ice
m.slr_gsic_equil_temp = Param(initialize=-1) # Equilibrium temperature
m.slr_cumgis = Var(m.t, within=NonNegativeReals, units=quant.unit("m"))
m.slr_gis_melt_rate_above_thresh = Param(
initialize=1.11860082
) # Melt rate above threshold
m.slr_gis_init_melt_rate = Param(initialize=0.6) # Initial melt rate
m.slr_gis_init_ice_vol = Param(initialize=7.3) # Initial ice volume
m.total_SLR = Var(m.t, within=NonNegativeReals, units=quant.unit("m"))
# Constraints relating to SLR
constraints = [
# Thermal expansion
GlobalConstraint(
lambda m, t: (
m.slr_thermal[t]
== slr_thermal_expansion(m.slr_thermal[t - 1], m.temperature[t - 1], m)
if t > 0
else Constraint.Skip
),
name="SLR_thermal",
),
GlobalInitConstraint(
lambda m: m.slr_thermal[0] == slr_thermal_expansion_init(m)
),
# GSIC
GlobalConstraint(
lambda m, t: (
m.slr_cumgsic[t]
== slr_gsic(m.slr_cumgsic[t - 1], m.temperature[t - 1], m)
if t > 0
else Constraint.Skip
),
name="SLR_GSIC",
),
GlobalInitConstraint(lambda m: m.slr_cumgsic[0] == 0.015),
# GIS
GlobalConstraint(
lambda m, t: (
m.slr_cumgis[t] == slr_gis(m.slr_cumgis[t - 1], m.temperature[t - 1], m)
if t > 0
else Constraint.Skip
),
name="SLR_GIS",
),
GlobalInitConstraint(lambda m: m.slr_cumgis[0] == 0.006),
# Total SLR is sum of each contributing factors
GlobalConstraint(
lambda m, t: m.total_SLR[t]
== m.slr_thermal[t] + m.slr_cumgsic[t] + m.slr_cumgis[t],
name="total_SLR",
),
]
return constraints
|