Appendix: WASP model equations
Description of the efficient earth system model
This Appendix describes the equations used in the new efficient Earth System Model; the Warming, Acidification and Sea-level Projector (WASP) (Fig. 2).
Anthropogenic ocean heat uptake in the WASP model
WASP assumes that for an imposed radiative forcing, R
TOTAL
(Wm−2), the eventual anthropogenic heat content increase for the ocean surface mixed layer at equilibrium, \( H_{equil}^{mixed} \) (J), is given via,
$$ {{H_{equil}^{mixed} } \mathord{\left/ {\vphantom {{H_{equil}^{mixed} } {V_{mixed} }}} \right. \kern-0pt} {V_{mixed} }} = c_{p} r_{SST:SAT} \lambda^{ - 1} R_{TOTAL} , $$
(3)
where r
SST:SAT
is the ratio of warming for global SSTs and surface atmosphere temperatures at equilibrium, c
p
is the mean specific heat capacity of seawater in the global ocean (Williams et al. 2012), and V
mixed
is the volume of the surface mixed layer (Fig. 2). To calculate the rate of anthropogenic heat uptake by the Earth System in WASP, N (W m−2), the radiative forcing, R
TOTAL
, is modulated by the fractional distance from the eventual anthropogenic heat uptake at equilibrium in the surface mixed layer at time t,
$$ N(t) = \left( {\frac{{H_{equil}^{mixed} (t) - H^{mixed} (t)}}{{H_{equil}^{mixed} (t)}}} \right)R_{TOTAL} (t), $$
(4)
where N for the whole Earth System is calculated by the distance from equilibrium heat uptake of the surface mixed layer. Equation (4) results in N(t) = R
TOTAL
(t) initially, at the moment that R
TOTAL
(t) is introduced and before the heat content of the mixed layer is increased, and N(t) = 0 at surface-ocean thermal equilibrium, once \( H^{mixed} = H_{equil}^{mixed} \). The anthropogenic heat uptake by ocean between times t − δt and t, δH (J), is given by,
$$ \delta H = f_{heat} A_{Earth} N\delta t, $$
(5)
where A
Earth
is the surface area of the entire planet Earth (see Appendix Table 5), and f
heat
is the fraction of whole Earth-System heat uptake within the ocean (Fig. 3f).
Table 5 Definitions of the fixed model parameters in WASP, and their values used in the model ensembles
The flux of anthropogenic heat content from the surface mixed layer into each sub-surface region between time t − δt and t, \( F_{Heat}^{mix \to region} \) (J), is calculated assuming that a passive tracer in the sub-surface ocean restores towards the mixed layer concentration according to defined e-folding timescales,
$$ \begin{aligned} F_{Heat}^{mix \to region} &= \left[ {H_{Usat}^{region} - r_{sub:SST} H_{Usat}^{mix} \left( {\frac{{V_{region} }}{{V_{mix} }}} \right)} \right]\exp \left( { - {{\delta t} \mathord{\left/ {\vphantom {{\delta t} {\tau_{region} }}} \right. \kern-0pt} {\tau_{region} }}} \right) \\ & \quad - \left[ {H_{Usat}^{region} - r_{sub:SST} H_{Usat}^{mix} \left( {\frac{{V_{region} }}{{V_{mix} }}} \right)} \right] \\ \end{aligned} $$
(6)
where τ
region
(yr) refers to the restoring timescale for tracers to equilibrate in a sub-surface ocean region of volume V
region
(m3) (where region refers to either the upper ocean, intermediate ocean, deep ocean or bottom ocean boxes, Fig. 2), H
region
is the anthropogenic heat content of a sub-surface ocean region, V
mix
(m3) is the volume of the surface mixed layer (see Appendix Table 5) and r
sub:SST
is the ratio of warming for the global sub-surface ocean to global SSTs at equilibrium.
The change in anthropogenic heat content in the surface mixed layer between time t − δt and t, δH
mixed (J), is then calculated from Eqs. (5) and (6), by,
$$ \delta H^{mixed} = A_{Earth} N(t)\delta t - F_{heat}^{mix \to upper} - F_{heat}^{mix \to inter} - F_{heat}^{mix \to deep} - F_{heat}^{mix \to bottom} , $$
(7)
while the changes in anthropogenic heat content in each sub-surface region, δH
region (J), is given by,
$$ \delta H^{region} = F_{heat}^{mix \to region} . $$
(8)
The ocean carbon cycle in the WASP model
The WASP model code considers separately the cumulative carbon emission that has remained in the air–sea system since the preindustrial at time t, \( I_{em}^{At + Oc} (t) \) (PgC), and the residual carbon uptake by the terrestrial system since the preindustrial, ΔI
ter
(t) (PgC), combining them together to find to total cumulative carbon emission, \( I_{em}^{TOTAL} (t), \)
$$ I_{em}^{TOTAL} (t) = I_{em}^{At + Oc} (t) +\Delta I_{ter} (t), $$
(9)
where details of how ΔI
ter
is calculated are given below in a later section of the Appendix.
In the timestep between times t − δt and t, CO2 is emitted into the air–sea system, \( \delta I_{em}^{At + Oc} \) (PgC), and is assumed to enter the atmosphere, increasing the undersaturation of carbon in the ocean (Goodwin et al. 2015) by δI
Usat
(emissions) (PgC). The increase in δI
Usat
(emissions) from \( \delta I_{em}^{At + Oc} (t) \) is calculated by re-arranging the time-dependent CO2 equation of Goodwin et al. (2015), \( \Delta \ln {\text{CO}}_{2} (t) = \left[ {I_{em}^{At + Oc} (t) + I_{Usat} (t)} \right]/I_{B} , \) and including an additional term for equivalent carbon emissions from the ocean temperature-CO2 solubility feedback (Goodwin and Lenton 2009). Applying to a single time-step gives,
$$ \delta I_{Usat} (emissions) = I_{B} \ln \left( {\frac{{I_{CO2} + \delta I_{em}^{At + Oc} + \delta I_{em}^{heat} }}{{I_{CO2} }}} \right) - \left( {\delta I_{em}^{At + Oc} + \delta I_{em}^{heat} } \right), $$
(10)
where \( I_{{{\text{CO}}_{2} }} \) is the amount of CO2 in the atmosphere in PgC, I
B
(PgC) is the buffered carbon inventory (Goodwin et al. 2007), and \( \delta I_{em}^{heat} \)(PgC) is the additional term representing the equivalent carbon emission from the temperature-solubility feedback between times t − δt and t. \( \delta I_{em}^{heat} \) is calculated using the methodology of Goodwin and Lenton (2009) using
$$ \delta I_{em}^{heat} = - \left( {\frac{{\partial C_{sat} }}{\partial T}} \right)\frac{{\delta H^{mix} }}{{c_{p} }}, $$
(11)
where ∂C
sat
/∂T is the sensitivity of DIC solubility to temperature, given in WASP by \( {{\partial C_{sat} } \mathord{\left/ {\vphantom {{\partial C_{sat} } {\partial T}}} \right. \kern-0pt} {\partial T}} = - 0.1 + 0.025\;\Delta \ln {\text{CO}}_{2} \) in g C m−3 K−1 (Goodwin and Lenton 2009; see Fig. 2 therein). The total \( \delta I_{Usat} (emissions) \) for the whole ocean (Eq. 10) is apportioned to the different ocean boxes (Fig. 2) according to their volume by,
$$ \delta I_{Usat}^{region} (emissions) = \left( {\frac{{V^{region} }}{{V^{total} }}} \right)\left[ {I_{B} \ln \left( {\frac{{I_{{{\text{CO}}_{2} }} + \delta I_{em}^{At + Oc} + \delta I_{em}^{heat} }}{{I_{{{\text{CO}}_{2} }} }}} \right) - \left( {\delta I_{em}^{At + Oc} + \delta I_{em}^{heat} } \right)} \right]. $$
(12)
The flux of carbon from the atmosphere to the surface mixed layer between times t − δt and t, \( F_{Usat}^{air \to mix} \) (PgC), acts to reduce mixed-layer I
Usat
over an e-folding timescale \( \tau_{mix} \)(yr), via,
$$ F_{Usat}^{air \to mix} = I_{Usat}^{mixed} (t - \delta t)\left( {1 - \exp \left( {\frac{ - \delta t}{{\tau_{mix} }}} \right)} \right). $$
(13)
The flux of I
Usat
exchanged from the mixed layer to each sub-surface ocean region between times t − δt and t, \( F_{Usat}^{mix \to region} \) (PgC), is calculated using the same restoring timescales as for the heat content fluxes, analogous to Eq. (6), via,
$$ F_{Usat}^{mix \to region} = \left[ {I_{Usat}^{region} - I_{Usat}^{mix} \left( {{{V^{region} } \mathord{\left/ {\vphantom {{V^{region} } {V^{mix} }}} \right. \kern-0pt} {V^{mix} }}} \right)} \right]\exp \left( { - {{\delta t} \mathord{\left/ {\vphantom {{\delta t} {\tau^{region} }}} \right. \kern-0pt} {\tau^{region} }}} \right) - \left[ {I_{Usat}^{region} - I_{Usat}^{mix} \left( {{{V^{region} } \mathord{\left/ {\vphantom {{V^{region} } {V^{mix} }}} \right. \kern-0pt} {V^{mix} }}} \right)} \right]. $$
(14)
Therefore, the overall change in I
Usat
in the mixed layer between t − δt and t, \( \delta I_{Usat}^{mix} \) (PgC), is given by,
$$ \delta I_{Usat}^{mix} = \delta I_{Usat}^{mix} (emissions) + F_{Usat}^{air \to mix} - F_{Usat}^{mix \to upper} - F_{Usat}^{mix \to inter} - F_{Usat}^{mix \to deep} - F_{Usat}^{mix \to bottom} , $$
(15)
while the changes in the sub-surface regions, \( \delta I_{Usat}^{region} \) (PgC), are given by,
$$ \delta I_{Usat}^{region} = \delta I_{Usat}^{region} (emissions) + F_{Usat}^{mix \to region} . $$
(16)
The sum of all changes to undersaturation carbon content in all timesteps since the preindustrial, and for all ocean regions, is then used to calculate the total cumulative ocean carbon undersaturation at time t, I
Usat
(t) (PgC), via,
$$ I_{Usat} (t) = I_{Usat}^{mix} (t) + I_{Usat}^{upper} (t) + I_{Usat}^{{\text{int} er}} (t) + I_{Usat}^{deep} (t) + I_{Usat}^{bottom} (t). $$
(17)
Atmospheric CO2 at time t, CO2 (ppm), is then calculated using the time-dependent CO2 relationship of Goodwin et al. (2015), with the additional term representing equivalent carbon emissions from the ocean temperature-CO2 solubility feedback,
$$ {\text{CO}}_{2} (t) = {\text{CO}}_{2} (t_{0} )\exp \left( {\frac{{I_{em}^{At + Oc} (t) + I_{Usat}^{total} (t) + I_{em}^{heat} (t)}}{{I_{B} }}} \right), $$
(18)
where t
0 refers to the preindustrial.
Sea-surface ocean acidification in WASP is calculated by using coefficients for the sensitivity of pH to Δln CO2 for seawater at chemical saturation with overlying atmospheric CO2, and for the sensitivity of pH to the distance of DIC from CO2 saturation in the mixed layer. This results in,
$$ \Delta pH(t) = c_{pH1}\Delta \ln {\text{CO}}_{2} (t) + c_{pH2} \left( {{{ - I_{Usat}^{mix} } \mathord{\left/ {\vphantom {{ - I_{Usat}^{mix} } {V^{mix} }}} \right. \kern-0pt} {V^{mix} }}} \right), $$
(19)
where the values c
pH1 and c
pH2 are calculated from perturbation experiments using an explicit numerical carbonate chemistry solver (Follows et al. 2006).
When carbon emissions are set to restore CO2 to a prescribed pathway, CO
restore2
(t) (Meinshausen et al. 2011), the carbon emissions added into the air–sea system between times t − δt and t, \( \delta I_{em}^{At + Oc} \), are set equal to the flux of undersaturation carbon from the air to the mixed layer flux from t − δt to t, plus 0.9 times the difference between the prescribes CO2 mixing ratio being restored towards at time t, and the actual model CO2 mixing ratio at t − δt,
$$ \delta I_{em}^{At + Oc} = F_{Usat}^{mix} + 0.9 \times \left( {{\text{CO}}_{2}^{restore} (t) - {\text{CO}}_{2} (t - \delta t)} \right), $$
(20)
where the factor of 0.9 is used to prevent CO2 from overshooting the restored value and causing a numerical instability whereby the simulated CO2 value oscillates around the prescribed pathway.
Anthropogenic surface warming in the WASP model
In WASP, the warming-CO2 emission relationship of Goodwin et al. (2015) is applied to calculate global mean surface warming, with additional terms for the radiative forcing from both Kyoto-protocol agents other than CO2, \( R_{{non{-}{\text{CO}}_{2} }} \) (W m−2), and from non-Kyoto agents, R
non-Kyoto
(Wm−2), and an additional term representing the equivalent cumulative carbon emissions from the ocean temperature-CO2 solubility feedback, \( I_{em}^{heat} \)(PgC) (Goodwin and Lenton 2009). The anthropogenic warming at time t, ΔT (K), is calculated via,
$$ \Delta T(t) = \left( {\frac{1}{\lambda }} \right)\left( {1 - \frac{\varepsilon N(t)}{R(t)}} \right)\left\{ {\left[ {\left( {\frac{a}{{I_{B} }}} \right)\left( {I_{Usat} (t) + I_{em}^{At + Oc} (t) + I_{em}^{heat} (t)} \right)} \right] + R_{{non{-}{\text{CO}}_{2} }} (t) + R_{non{-}Kyoto} (t)} \right\}. $$
(21)
where ε is the efficacy of ocean heat uptake, a = 5.35 Wm−2 is the radiative forcing coefficient from a log change in CO2 (Myhre et al. 1998), and \( R_{{non{-}{\text{CO}}_{2} }} \) is an input parameter from the RCP scenario definitions (Meinshausen et al. 2011), and R
non-Kyoto
is scaled from \( R_{{non{-}{\text{CO}}_{2} }} \) after Eq. (2) to capture uncertainty in anthropogenic radiative forcing.
WASP can also simulate the thermosteric and isostatic contributions to global mean sea level rise using the internally simulated ocean heat uptake and surface warming. The thermosteric contribution to sea level rise can be estimated as a function of ocean heat uptake after Williams et al. (2012; see Eq. 4 therein), while the semi-empirical approach of Rahmstorf (2007) can be applied for isostatic sea level rise (Rahmstorf 2007; see Eq. 1 therein).
Terrestrial carbon cycling in the WASP model
Terrestrial carbon is represented by soil carbon and vegetation carbon reservoirs (Fig. 2). The rates of change of vegetation carbon, I
veg
(PgC) and soil carbon, I
soil
(PgC), reservoirs at time t are modelled by,
$$ \frac{{dI_{Veg} }}{dt}(t) = {\text{NPP}}(t_{0} )f_{1} (\Delta T)f_{2} ({\text{CO}}_{2} ) - LL(t_{0} )\frac{{I_{Veg} (t)}}{{I_{Veg} (t_{0} )}}, $$
(22)
and,
$$ \frac{{dI_{Soil} }}{dt}(t) = {\text{LL}}(t_{0} )\frac{{I_{Veg} (t)}}{{I_{Veg} (t_{0} )}} - {\text{SR}}(t_{0} )\frac{{I_{Soil} (t)}}{{I_{Soil} (t_{0} )}}f_{3} (\Delta T), $$
(23)
where NPP (PgC yr−1) is net primary production, LL (PgC yr−1) is leaf litter fallout and SR (PgC yr−1) is soil respiration (Fig. 2), and t
0 refers to the preindustrial. At preindustrial steady state, t = t
0 the system is in balance, giving,
$$ {\text{NPP}}(t_{0} ) = {\text{LL}}(t_{0} ) = {\text{SR}}(t_{0} ). $$
(24)
The functions f
1–f
3 must return unity at preindustrial CO2 and ΔT = 0. A simple linear response of global NPP to global mean surface warming is assumed, hiding a multitude of mechanistic factors, giving,
$$ f_{1} (\Delta T) = 1 + \frac{{\left( {{{\partial {\text{NPP}}} \mathord{\left/ {\vphantom {{\partial {\text{NPP}}} {\partial T}}} \right. \kern-0pt} {\partial T}}} \right)}}{{{\text{NPP}}(t_{0} )}}\Delta T. $$
(25)
The response of global NPP to CO2 is modelled by an empirical CO2-fertilisation effect related to the log-change in atmospheric CO2 (Alexandrov et al. 2003), resulting in a function f
2 in (22) of,
$$ f_{2} ({\text{CO}}_{2} ) = 1.0 + \gamma_{{{\text{CO}}_{2} }}\Delta \ln {\text{CO}}_{2} , $$
(26)
where \( \gamma_{{{\text{CO}}_{2} }} \) is the Keeling-equation CO2-fertilisation growth factor (Alexandrov et al. 2003). Global mean soil carbon residence time is assumed to be linearly related to warming, ΔT, via a specified sensitivity, ∂τ/∂T (yr K−1), using,
$$ f_{3} (\Delta T) = \frac{{\tau (t_{0} )}}{\tau (t)} = \frac{{\left( {{{I_{Soil} (t_{0} )} \mathord{\left/ {\vphantom {{I_{Soil} (t_{0} )} {{\text{NPP}}(t_{0} )}}} \right. \kern-0pt} {{\text{NPP}}(t_{0} )}}} \right)}}{{\left( {{{I_{Soil} (t_{0} )} \mathord{\left/ {\vphantom {{I_{Soil} (t_{0} )} {{\text{NPP}}(t_{0} )}}} \right. \kern-0pt} {{\text{NPP}}(t_{0} )}}} \right) + \left( {{{\partial \tau } \mathord{\left/ {\vphantom {{\partial \tau } {\partial T}}} \right. \kern-0pt} {\partial T}}} \right)\Delta T}} $$
(27)
The cumulative changes in I
soil
and I
veg
over time since the preindustrial, ΔI
soil
and ΔI
veg
respectively, are calculated and summed to give the total change in terrestrial carbon since the preindustrial,
$$ \Delta I_{ter} =\Delta I_{soil} +\Delta I_{veg} , $$
(28)
and combined with the cumulative emission to the air–sea system, \( I_{em}^{At + Oc} \), to give the total compatible carbon emission using Eq. (9).