# A continuous latitudinal energy balance model to explore non-uniform climate engineering strategies

## Abstract

In order to investigate the effects of solar radiation management (SRM) technologies for climate engineering, an analytical model describing the main latitudinal dynamics of the Earth’s climate with closed-loop control has been developed. The model is a time-dependent Energy Balance Model (EBM) with latitudinal resolution and allows for the evaluation of non-uniform climate engineering strategies. The resulting partial differential equation is solved using a Green’s function approach. This model offers an efficient analytical approach to design strategies that counteract climate change on a latitudinal basis to overcome regional disparities in cooling. Multi-objective analyses are considered and time-dependent analytical expressions of control functions with latitudinal resolution can be obtained in several circumstances. Results broadly comparable with the literature are found, demonstrating the utility of the model in rapidly assessing new climate engineering controls laws and strategies. For example, the model is also used to quickly assess the trade-off between the number of degrees of freedom of SRM and the rms error in latitudinal temperature compensation. Moreover, using the EBM the dynamics of the ice line can be investigated and a Lyapunov stability analysis is employed to estimate the maximum reduction of solar insolation through climate engineering before the current climate falls into an ice-covered state. This provides an extreme operational boundary to future climate engineering ventures.

## Keywords

SRM Climate engineering Climate modelling Multi-objective control Ice line Lyapunov stability## 1 Introduction

Climate engineering Blackstock et al. (2009) aims to partly offset the impacts of human-driven climate change. It involves techniques developed both to reduce the concentration of carbon dioxide in the atmosphere [Carbon Dioxide Removal (CDR) methods] and to counteract the radiative forcing generated [Solar Radiation Management (SRM) methods] Vaughan and Lenton (2011). High-speed computers and high-fidelity numerical models for the climate system can be used to evaluate climate engineering strategies. For example, General Circulation Models (GCMs) have many millions of degrees of freedom and, although they can be physically realistic, these models are computationally expensive. In contrast, low order one-dimensional climate models can be found in Budyko (1969), Sellers (1969) and North et al. (1981). In particular, considering elementary thermodynamics, Budyko (1969) and Sellers (1969) proposed low order climate models to investigate the climate state as a function of the solar constant, whereas North et al. (1981) investigated in detail global energy balance models using a general transport term. Moreover, a Green’s function approach is used in North et al. (1981) to obtain the explicit analytical solution of a diffusive climate model in terms of hypergeometric functions.

This paper aims to approximate the complexity of the climate system with a simple model that takes into account its main features as described in North et al. (1981), and to develop continuous control laws for climate engineering as a function of latitude. Therefore, efforts have been made to develop a continuous-time PDE system with latitudinal resolution with which it is possible to explore control strategies to begin to investigate issues related to regional disparities and the side effects of SRM techniques. This continuous PDE model extends a simple 3 box model which has been used to investigate the use of adaptive control for climate engineering Bonetti and McInnes (2018).

The use of efficient analytic methods provides a useful tool to rapidly asses SRM strategies with latitudinal resolution and allows efficient application of multi-objective analyses. Moreover, through this procedure the latitudinal impact of SRM can be directly addressed. In general, the pattern of insolation reduction that can be generated using SRM does not match the pattern of climate change impacts due to increased forcing due to \(\text {CO}_2\). In the literature regional disparities due to SRM have been widely discussed (Heyen et al. 2015; Moreno-Cruz et al. 2012). In MacMartin et al. (2012) and Kravitz et al. (2016) an atmosphere-ocean general circulation model (AOGCM) is used to explore the potential of SRM with multiple degrees of freedom. Also, in [9], Kravitz et al. (2017), MacMartin et al. (2017), injections of sulphate aerosols at multiple locations are considered to counteract incoming solar radiation with a coupled atmosphere-ocean general circulation model.

Despite that the analytical model developed in this paper is a simplification of other more realistic numerical models, broadly comparable results are found. This demonstrates the utility of the model in rapidly assessing new climate engineering strategies and controls laws. Again, the model can assess the trade-off between the number of degrees of freedom of SRM and the RMS error in latitudinal temperature compensation, for example.

Constraints on SRM are also explored through ice line dynamics, providing extreme operational limits on SRM obtained through the latitudinal model. This analysis illustrates the scale of the required SRM manipulation and insolation reduction that would be required to trigger an instability of the climate system, demonstrating that this limit is far beyond expected SRM interventions. With respect to other similar calculations in the literature (Coakley 1979; North et al. 1981; Schneider and Gal-Chen 1973), the insolation reduction that would trigger climate instability is given as a function of both latitude and time, providing more insight for appropriate comparisons with commonly considered SRM strategies. Moreover, the time-dependence of the model employed, and of the control function for insolation reduction, produces results which are less sensitive to changes in the solar constant Schneider and Gal-Chen (1973). This is a fundamental issue with the dynamics of the ice line and for SRM investigations in general.

Section 2.1 addresses a mathematical model to describe the climate system with a single partial differential equation (PDE); here, the analytical solution to compute the temperature perturbation due to atmospheric carbon dioxide (\(\text {CO}_2\)) is developed. Then, considering the deployment of climate engineering through a reduction of incoming solar radiation, a control law to drive the temperature perturbation to zero is developed. In Sect. 3 a multi-objective control analysis is undertaken with a PI feedback control. In Sect. 4 the PDE model is employed to find an analytical control law to achieve a desired temperature profile under a doubling of \(\text {CO}_2\) and in Sect. 5 constraints on SRM are investigated by exploring the effect of climate engineering on the dynamics of the ice line.

## 2 Model and methodology

### 2.1 PDE model for the climate system

In this section a dynamical model of the climate system with latitudinal resolution is developed. With respect to North et al. (1981), where a one-dimensional equilibrium model with diffusive heat transfer was developed to investigate ice feedback mechanisms, here temperature is also considered as a function of time. Moreover, both hemispheres are considered and differences in land and oceans are taken into account for the computation of the heat capacity for the northern and southern hemispheres. The model is then employed to explore control strategies based on a reduction of incoming solar radiation.

In general, Eq. (1) allows for the computation of the zonally-averaged surface temperature *T*, as function of the time *t* and the sine of latitude \(x=\text {sin}(\phi )\). The model allows for a range of forcing terms, therefore the presence of \(\text {CO}_2\) forcing and a control function representing the deployment of SRM strategies will be included later. The solution is constrained by boundary conditions, since the heat flux must vanish at the poles and only solutions with no heat transport across the equator are considered.

Increasing the forcing in one hemisphere relative to the other causes a shift of the latitude of zero heat-flux inducing large precipitation anomalies. However, this effect is not taken into account in order to keep the analysis manageable.

Equation (1) represents an EBM where the incoming and outgoing energy are balanced and an equilibrium temperature distribution with latitude is reached. Specifically, *T*(*x*, *t*) represents the annual zonally-averaged temperature field. The incoming energy is the solar radiation, and the energy losses are given by the effect of the Earth’s albedo and the infra-red radiation leaving the top of a latitudinal element. The energy transported by a latitudinal element to its neighbours due to the movement of geophysical fluids is represented by a diffusion process where the transport is proportional to the gradient of the temperature field. As is shown later, some of these constants are taken from data in the literature (North et al. 1981; McGuffie and Henderson-Sellers 2005; Budyko 1969) and others are chosen in order to match the time-domain step response of high-fidelity numerical models. In this way it is possible to regulate the equilibrium climate sensitivity of the system which is a key parameter for comparison between climate models. Therefore, the goal of this model is emulating the behaviour of complex numerical models with a more convenient analytical structure to easily (and rapidly) implement climate engineering control strategies based on SRM.

*A*and

*B*are empirical constants selected to account for the effect of clouds, water vapour and \(\text {CO}_2\). In particular, in Budyko (1969) an infra-red parametrization for the northern and southern hemispheres is found and \(I_{IR}\) can be written as:

*S*(

*x*) describes the distribution of the incident solar radiation averaged over 1 year for which the expression used in North (1975) is considered: \(S(x)=1+S_2P_2(x)\) where \(S_2=-0.477\) is a constant and \(P_2(x)=\frac{1}{2}\left( 3x^2-1 \right) \) is the second Legendre polynomial. Then, \(\alpha (x,x_s)\) is the planetary co-albedo at latitude

*x*North et al. (1981) for which a smooth albedo formulation, that includes the definition of the ice line at \(x=x_s\), is considered Widiasih (2013):

*M*represents the steepness of the albedo function near the ice line and is set to 12. The value of \(x_s\) for the current climate is set to \(x_{s0}=\pm \, 0.95\) for the northern and southern hemisphere, respectively. Moreover, in Eq. (1)

*D*is an empirical constant describing the latitudinal transport of energy. Its value for the northern hemisphere is given by \(0.649 \ {\text {W}}{/}{\text {m}}^2\,{/}^{\circ } \text {C}\)North et al. (1981), whereas the value for the southern hemisphere is found in order to satisfy the condition \(T_N(0,\infty )=T_S(0,\infty )\), where \(T_{N/S}(x,\infty )\) are the temperature fields at latitude

*x*for the northern and southern hemispheres in the equilibrium state (\(t\rightarrow \infty \)). The value of

*D*for the southern hemisphere that satisfies this condition is \(0.73 \ {\text {W}}{/}{\text {m}}^2\,{/}^{\circ } {\text {C}}\).

Finally, in Eq. (1) *C* is the effective heat capacity, which is largely determined by the oceans. The values of the heat capacities are estimated for the northern and southern hemispheres considering the different hemispherical distributions of land and water. In particular, a larger fraction of water can be found in the southern hemisphere and this leads to a larger heat capacity. The heat capacity over land is approximately 1/30 of the capacity over the ocean mixed layer North et al. (1981). Therefore considering the fraction of water and land in each hemisphere (oceans cover the 61% of the northern hemisphere and the 82% of the southern hemisphere) the heat capacity, in terms of *B*, is 2.88*B* years for the northern hemisphere and 3.79*B* years for the southern hemisphere.

It is worth noting that the model neglects the mean circulation in both the atmosphere and oceans but includes heat transport due to circulation. This approach allows for tractable mathematics and analytical solutions.

The system in Eq. (1) with the boundary conditions in Eq. (2) can be identified as the non-homogeneous heat equation with Neumann boundary conditions and has an analytical solution. An efficient, straightforward approach for solving such problems and obtaining an analytical solution is provided by the Green’s function formalism. Green’s functions are constructed by utilizing the eigenfunctions and eigenvalues of the differential operators from which the system is constructed Cole et al. (2010). Once the Green’s function for a given problem is known, the solution for the latitudinal distribution of the temperature is immediately computed from the analytical expression for the Green’s function Hahn and Ozisik (2012).

*P*(

*n*,

*x*) is the nth-degree Legendre Polynomial computed in

*x*. The Green’s function \(G(x,\xi ,t,\tau )\) represents the temperature perturbation \(\delta T(x,t)\), at latitude

*x*, at time

*t*, due to an instantaneous heat source of unit strength, located at \(\xi \), releasing its energy instantaneously at time \(\tau \). Therefore, the argument ‘\(\xi ,\tau \)’ in Eq. (6) represents the impulse, given by the heat source term Hahn and Ozisik (2012), whereas ‘

*x*,

*t*’ represent the resulting effect. Moreover, \(\varPhi \) is the external forcing of the system which in this case depends only on

*x*and can be written as:

*T*(

*x*, 0) and integrated over the hemisphere. Whereas, the second term represents the contribution of the forcing to the temperature profile. Then

*T*(

*x*, 0) is the latitudinal distribution of the temperature (at \(t=0\)) which can be found through a similar procedure considering the 1D-model. Its analytical expression is given by:

*T*(

*x*,

*t*) in Eq. (5) at the final time \(t=t_f\). The result is reported in Fig. 1 and is consistent with the literature (North et al. 1981; Budyko 1969; Schneider and Gal-Chen 1973). Considering the combination of the responses in the two hemispheres, the step response of the PDE model for the climate system can be found in Fig. 2 together with the response from a first order linear model (dashed line) Kravitz et al. (2016) and a semi-infinite diffusion model MacMynowski et al. (2011). The behaviour of the PDE model is comparable with the semi-infinite diffusion model. In particular, the two curves (continuous black thick and thin lines) reach the same equilibrium value, although the relaxation profile is different. In the semi-infinite diffusion model, the overall heat capacity is given by 4.063 years with an equilibrium climate sensitivity of \(2.71\,^{\circ } \text {C}\) MacMynowski et al. (2011), which is comparable with the HadCM3L model. As can be seen in Fig. 2, in this case, the perturbation is considered with respect to the equilibrium temperature. This version of the model, where only the temperature anomaly with respect to the equilibrium state is considered, is described in detail in the next section. According to Robert and North (1979) and North et al. (1981), the fundamental sensitivity of the system can be estimated for the PDE model as follows:

## 3 Multi-objective control strategies

It is important to highlight that the expression for the external forcing \(\varPhi (x,t)\) is a generic function; therefore, it can be used to implement climate engineering strategies and analyse the behaviour of the PDE model. In this section a slightly modified version of the model in Eq. (1), where the system is considered in the neighbourhood of its equilibrium state, is presented.

This version includes external forcing which represents the excess of carbon dioxide in the atmosphere. This forcing term (\(F_{\text {CO}_2}\)) causes an imbalance in the radiative forcing in Eq. (1) producing an anomaly in the latitudinal temperature profile. In this model, the anomaly indicates exclusively the effect of an excess of \(\text {CO}_2\) in the atmosphere. In particular, the 1pct\(\text {CO}_2\) scenario Collins et al. (2013), where a constant increase in carbon dioxide concentration of \(1\%\) per year is assumed, is considered in this section.

Assuming the deployment of a climate engineering strategy consisting of a reduction of the incoming solar radiation (SRM), a control function, defined as \(U(x,t)Q_0\) \(S(x)\alpha (x,x_s)\), which in general is a function of both latitude and time, will represent the climate engineering intervention in terms of the fractional reduction of the incoming solar radiation. This term is included in the model through Eqs. (10–11) and aims to reduce the temperature anomaly generated by the excess of atmospheric \(\text {CO}_2\).

*T*(

*x*, 0) is equal to zero and only the perturbative term of Eq. (5) is considered in Eq. (10).

The analytical expression for the temperature distribution, obtained through the Green’s function approach, allows for a fast and efficient investigation of the effects of SRM deployment on the climate system. The advantages of using such an approximate mathematical model for closed-loop control purposes can be summarised as follow: (1) capturing latitudinal disparities in induced cooling; (2) easy application of optimization processes and multi-objective analyses; (3) clearer understanding of the key climatic processes involved and the effects of closed-loop control on them; (4) the possibility of developing an analytical control function with latitudinal resolution; (5) the efficient assessment of new climate engineering strategies, prior to more detailed analysis using GCMs.

*cc*is the current \(\text {CO}_2\) concentration in the atmosphere [400 ppm Dlugokencky and Tans (2016)], \(cc_0\) is the pre-industrial level of carbon dioxide [278 ppm Dlugokencky and Tans (2016)] and \(\lambda = \log (1.01)\) represents the \(1 \%\) per year growth rate.

*f*represent the extremes of the hemispheric integration.

As in Kravitz et al. (2016), the first goal will now consider the minimization of the global mean temperature \(\delta T_0(t)\) only (case 1), the second case also considers the minimization of the temperature gradient \(\delta T_1(t)\) (case 2), and the third case investigates the full problem where all the three objectives are taken into account (case 3).

A Proportional-Integral (PI) control is employed to achieve the required control objectives. This control structure is a feedback control strategy, where the extent of the control at time *t* depends on the state \(\delta T_i\) at the previous time \((t-1)\) (years). This approach is justified considering a reasonable time delay for the deployment of SRM strategies. The observations of the temperature distribution during year *t* would be used to estimate the quantity of material required, for example if stratospheric aerosol injection is used. The time to collect data, implement decision-making processes are assumed to cause a delay between the time of observation and the deployment of the climate engineering strategy. In this paper a time-delay of 1 year is considered. Other important properties of feedback control systems and advantages of their use in climate engineering can be found in MacMartin et al. (2014) and MacMartin et al. (2014).

### 3.1 Results

Considering the analytical solution of the PDE system in Eq. (10), the three outputs, \(\delta T_0(t)\), \(\delta T_1(t)\) and \(\delta T_2(t)\), can be computed through Eq. (16). As noted earlier, climate engineering through the reduction of incoming solar radiation is deployed in case (1) to minimize the global mean temperature, in case (2) to drive both \(\delta T_0(t)\) and the temperature gradient \(\delta T_1(t)\) to zero and, finally, in case (3), all three outputs are controlled. Again, in this analysis the 1pct\(\text {CO}_2\) scenario is considered. To achieve these strategies, the control functions reported in Eqs. (18–20) are employed.

Summary of the control strategies considered

case | \(\delta T_0(t)\) | \(\delta T_1(t)\) | \(\delta T_2(t)\) | Control function |
---|---|---|---|---|

1 | \(\star \) | – | – | \(U_{1x1}(t)\) |

2 | \(\star \) | \(\star \) | – | \(U_{2x2}(x,t)\) |

3 | \(\star \) | \(\star \) | \(\star \) | \(U_{3x3}(x,t)\) |

The PI-control scheme is now fully defined and the control functions, in terms of the reduction of insolation, obtained. Figure 3a shows the time-history of \(U_{1x1}(t)\), which has a uniform distribution at every latitude since the first Legendre polynomial \(L_0\) does not depend on *x*. This strategy shows the effect of a latitudinally-uniform reduction of insolation that increases with time. However, although the increase of atmospheric \(\text {CO}_2\) is uniformly distributed there is an amplified effect at the poles. Therefore, when uniform cooling is applied as in case (1), an overcooling of the tropics and an undercooling of the poles occurs with the northern hemisphere cooler than the southern hemisphere [see also Kravitz et al. (2016)]. This result can be seen from Fig. 5 where the latitudinal distribution of the zonal mean temperature at the final time is reported for the three cases.

Hemispheric differences are related to the different distribution of ocean and land between the two hemispheres and, in particular, to the impact of the ocean on heat transport Kang and Seager (2012). These effects are taken into account in the PDE model through the values of the heat capacities and the transport coefficients employed for the northern and southern hemispheres.

Figure 3b shows the function \(U_{2x2}(x,t)\), employed in the case (2), which is given by the combination of the feedback control of \(\delta T_0(t)\) and \(\delta T_1(t)\). Its latitudinal distribution is dictated by the first and the second Legendre polynomial expressions in order to minimize the global mean temperature as well as the temperature gradient.

The additional feedback control of the inter-hemispheric temperature gradient in case (2) reduces disparities between the temperature residuals in the north pole and the south pole and decreases over-cooling of the tropics. Therefore, as can be seen in Fig. 3b, a larger cooling effect is required in the southern hemisphere.

As noted in Kravitz et al. (2016), the additional feedback control of the inter-hemispheric temperature gradient in case (3) reduces disparities between the temperature residuals in the north pole and the south pole and decreases over-cooling of the tropics. Again, these effects are also confirmed in Fig. 5.

Finally, Fig. 3c shows the distribution of the control function employed in the full case (3), where the equator-to-pole temperature gradient is also minimized.

In all three cases the control strategy employed is consistently comparable with the numerical results obtained in Kravitz et al. (2016), where multi-objective control strategies are applied to two fully coupled atmosphere-ocean general circulation models (AOGCM) that participated in CMIP5, the CESM 1.0.2 (Community Earth System Model) and the GISS ModelE2 (the Goddard Institute for Space Studies) in order to minimize \(\delta T_0(t)\), \(\delta T_1(t)\) and \(\delta T_2(t)\). In particular, the control responses obtained with the analytical solution of the PDE model are comparable with those obtained through the CESM. However the model shows low sensitivity to the control of \(L_1\) as occurs for the GISS model.

The results regarding the three outputs of the multi-objective control strategy simulations are reported in Fig. 4a–c. In particular, the output time-history of \(\delta T_0(t)\) is reported in Fig. 4a, whereas Fig. 4b, c show \(\delta T_1(t)\) and \(\delta T_2(t)\), respectively. Each figure includes three curves: the black line represents case (1), the dark-grey line represents case (2) and the light-grey line shows the full case (3). Therefore, it is possible to analyse the effect of every control strategy on each output. Moreover, a Gaussian noise (zero mean and standard deviation set to \(10^{-2}\)) is added to the outputs of the temperature level to simulate measurement noise and climate variability.

It can be seen from Fig. 4a–c that, in all the cases considered, the objective of each specific requirement is achieved. In fact the global mean temperature in case (1), the inter-hemispheric temperature gradient in case (2) and the equator-to-pole temperature gradient in case (3) are minimized. Although, from Fig. 4b, it can be noted that the system is not very sensitive to \(U_1\) and it is found that \(\delta T_1\) is reduced to a mean value of approximately 0.02 \(^{\circ } \text {C}\). This result is comparable with the outcome from the GISS model found in Kravitz et al. (2016). Moreover, negative effects are found for the objectives that are not managed in a particular case, such as \(\delta T_1\) in cases (1) and (3) and \(\delta T_2\) in cases (1) and (2).

In accordance with results from the literature Kravitz et al. (2016); Kravitz et al. (2011), the required latitudinally-uniform reduction of insolation increases linearly with time as the atmospheric \(\text {CO}_2\) concentration grows (see Fig. 3a) and mainly aims to decrease the global mean temperature \(\delta T_0\).

Finally, in a rather similar way to the literature the equator-to-pole temperature gradient (\(\delta T_2\)) in Fig. 4c shows convergence to zero steady-state error in case (3), while in case (1) shows large sensitivity to the climate variability (noise). It is therefore clear that the analytic PDE model can provide an efficient and effective means of investigating non-uniform climate engineering strategies.

Moreover, the model described in this paper allows also for the analysis of the zonal mean temperature. Therefore, a trade-off between the number of controlled degrees of freedom of SRM and the compensation of the zonal mean temperature is performed. Considering Eq. (10), the zonal mean temperature anomaly is computed and it is found that the rms error in compensating \(\delta T\) is \(0.84\,^{\circ } \text {C}\), \(0.81\,^{\circ } \text {C}\) and \(0.31\,^{\circ } \text {C}\) when \(U_{1x1}\), \(U_{2x2}\) and \(U_{3x3}\) are applied to the system respectively. These results are shown in Fig. 5, where the zonal mean temperature at final time is reported with latitude for the three cases considered. As can be seen, the temperature anomaly is noticeably lower in case (3). This result demonstrates that the zonal mean temperature is not completely minimized in any case, but that the overall rms error decreases when more degrees of freedom are managed. In fact, in the case when SRM is not deployed (\(U=0\)) the overall rms error due to the 1pct\(\text {CO}_2\) scenario is \(1.41\,^{\circ } \text {C}\). Thus, the computed control functions are able to manage the reduction of the global mean temperature anomaly, the temperature gradient and equator-to-pole temperature gradient, and greater benefits are found for the zonal mean temperature in all the three cases. In particular, case 3 is the most advantageous and indicates larger residuals of the zonal mean temperature anomaly when 3 degrees of freedom are considered.

## 4 Design of a control law to counteract a doubling of \(\text {CO}_2\)

The analytical solution of the PDE model reported in Eq. (5) can be employed to find a control law with latitudinal resolution to obtain a desired temperature profile. As demonstrated in the previous section, the PI-control is used in a closed-loop scheme to achieve the minimization of the required objectives. The final outcome is the appropriate time-dependent control function that takes into account latitudinal disparities.

In this section, a more generic case is investigated. The method required to find a control function to achieve a specific temperature profile is presented. Assuming a doubling of \(\text {CO}_2\) (which amounts to a forcing of \(F_{2 \times \text {CO}_2}=3.71 \ \text {W}/\text {m}^2\)) the necessary reduction of solar insolation to drive the temperature to the pre-industrial profile is evaluated and expressed through a control law with latitudinal resolution.

*U*(

*x*,

*t*), required to counteract the effect of \(F_{2 \times \text {CO}_2}\), can now be found by setting \(\varPhi (x,t)\) to zero.

In the second case (Fig. 8b), a constant radiative forcing equal to a doubling of \(\text {CO}_2\) is assumed and a slightly larger control effort is necessary overall. In fact the maximum value of *U*(*x*, *t*) required is \(4.5\%\) of the incoming solar radiation. This is due to the larger temperature anomaly caused by the steady radiative forcing with respect to the gradual change investigated in the first case. Moreover, integrating *U*(*x*, *t*) over latitude, it is possible to estimate the global mean solar insolation required to counteract a doubling of \(\text {CO}_2\). This is found to be \(1.78 \ \%\), which is comparable with the value of \(1.8\%\) found in literature where the global temperature is investigated for a doubling of the atmospheric \(\text {CO}_2\) content, such as in Bala et al. (2008) and Govindasamy and Caldeira (2000).

The increase in atmospheric \(\text {CO}_2\) concentration causes warming everywhere but requires a larger cooling at the poles. This can be justified considering the pattern of the incoming solar radiation (see Fig. 7). When SRM is considered, the latitudinal distribution of the control law is always related to the pattern of insolation and the response of the climate system, which in this paper is given by the PDE model described in Sect. 2.1.

*U*(

*x*,

*t*) is required to have an inverse latitudinal distribution with respect to the incoming solar radiation. Therefore, it is found that the required control is larger at the poles than at the equator. This result is again widely comparable with the literature, for example (Bala et al. 2008; Ban-Weiss andCaldeira 2010; Govindasamy et al. 2003).

## 5 Constraints on climate engineering

In this section the dynamics of the ice line is investigated using the analytical solution of the PDE model in Eq. (1). Climate engineering involves the manipulation of the climate system, therefore the analysis of the stability of the global climate (related to ice line dynamics) is of critical relevance for the study of the impact of climate engineering interventions.

This analysis is of key importance for climate engineering involving SRM since it demonstrates that the extent of the insolation reduction commonly considered for SRM is far from triggering large-scale instability of the climate system. The definition of safe operating boundaries is a requirement for engineering ventures. It will be shown that the PDE can be readily adapted for such analysis.

The Lyapunov stability criterion is now exploited to find the critical climate engineering intervention that would lead the current climate towards an ice-covered state. An upper limit on SRM is therefore found by exploring the effect of climate engineering on the dynamics of the ice line. While this limit is of course highly unlikely to be reached, the analytical PDE model provides insight into the extreme operational boundaries of SRM.

*x*at equilibrium, i.e. for \(t\rightarrow \infty \). Because of the ice line condition, Eq. (22) can also be written as \(T(x_s+\delta x_s,t)=T_s=T(x_s,\infty )\), therefore the variation of the ice line latitude is given by Robert and North (1979):

*U*. For the equator and the pole, respectively, the limits are set as \(x_0=0\) and \(x_1=1\). Moreover, \(G_Z(x,\xi )\) represents the temperature response to a ring of heat added/subtracted at a given latitude and includes the effect of the ice line shift to first order, as found in North et al. (1981).

*T*(

*x*,

*t*) under a perturbation \(\delta T\). In order to understand the meaning of \(\lambda \) it is necessary to report some results from the study of the linear stability of the 1-D model in North et al. (1981). In this study a transcendental equation, which is satisfied only for the stability eigenvalues \(\lambda \), is developed and can be written as follow:

*B*are given in Sect. 2.1 and the function \(F_{\lambda }\) is given by the right-hand side of Eq. (25). According to North et al. (1981), one way to obtain \(dQ/ dx_s\) is a relationship depending on the climate sensitivity of the model \(\left( \beta _0\right) \) given by:

The three equilibrium states are shown in Fig. 10, where the potential function \(\varPsi \) normalized over \(Q_0\) is illustrated as a function of \(T_0\). Here, \(T_0\) is the global equilibrium temperature, which is defined as the integral of *T*(*x*, 0) (see Eq. 8) with respect to latitude.

Solving Eq. (25) for \(\lambda \) with \(\frac{dQ}{dx_s}<0\), the lowest root is found to be \(\lambda =-0.3086\). As will be shown shortly, the value found for \(\lambda \) determines the reduction of insolation radiation required to drive the climate system from state I to II. Since state II is unstable (as can be seen in Fig. 10) any larger reduction of insolation would make the system fall into state III; therefore, the estimated insolation reduction represents the reduction of insolation required before an ice-covered state is achieved.

In Fig. 11 the lowest root of Eq. (25) is reported for several climate sensitivities and it can be seen that the greater the sensitivity of the model, the closer to zero the value of \(\lambda \). The PDE model has a sensitivity of \(2.74\,^{\circ } \text {C}\) for the northern hemisphere, again with \(\lambda =-0.3086\). In other cases, \(\lambda \) can be less than \(-1\) if the climate model has a sensitivity of \(2\,^{\circ } \text {C}\). Thus, as expected, in a more sensitive model a smaller change in insolation is necessary to reach instability.

Thus, a value of \(\lambda \) equal to \(-0.3086\) is now employed to find the control function which would trigger the climate instability and so the solution *T*(*x*, *t*) approaches the ice-covered stable solution. Therefore, the result of this investigation provides the limit of SRM in terms of the maximum reduction of insolation before the Earth’s climate approaches a new ice-covered state. Again, this provides an extreme operational boundary for SRM, which can be obtained from the analytic PDE model developed in Sect. 2.1. Again, it is clearly unlikely that such a control boundary would be reached.

*L*where it is not defined North et al. (1981):

*U*is obtained.

- (a)
\(v(x_{s0})=0\)

- (b)
\(v(x_s(x,t,U))>0\) for \(x_s\ne x_{s0}\)

- (c)
\(\frac{\partial }{\partial t}v(x_s(x,t,U))> 0\)

*V*is developed. Substituting Eq. (23) in Eq. (31), the analytical expression for \(V(x_s(x,t,U))\) can be developed as follow:

*U*, and solving the inequality \(\frac{\partial }{\partial t}V(x_s(x,t,U))> 0\) for the variable

*U*the following condition is obtained:

As can be seen from Eq. (34), condition (c) of the Lyapunov stability criterion is satisfied by an infinite number of control functions. Each of these functions is identified by a specific value of the parameter \(C_1\), which enables the selection of the initial condition for the control function *U*. In Fig. 12, control functions with values of \(C_1\) between 0 and 3 can be found.

As seen in Fig. 8, in order to counteract a doubling of \(\text {CO}_2\), a reduction of insolation of \(4-4.5 \%\) is required at the poles and only \(1-1.5 \%\) at the equator. Comparing this result with Fig. 13, although the cooling required at the equator to achieve the ice-covered state is considerably higher than \(1.5 \%\), the required deployment of SRM at the poles to counteract \(F_{2x\text {CO}_2}\) is in principle sufficient to move the ice line to lower latitudes. However, if the energy input to the tropics is left nearly constant, as in this case, changes in the albedo of the middle and upper latitudes can eventually be mitigated by the exporting of energy from the tropics Schneider and Gal-Chen (1973). Therefore, the instability of the system would not be triggered. This analysis is of importance to climate engineering involving SRM because it demonstrates that the extent of the insolation reduction commonly considered for SRM is rather far from such a catastrophic boundary.

*U*is substituted from \(U_\text {bound}\) and \(G_t\) is reported in Eq. (6) in Sect. 2.1.

*U*is substituted from \(U_\text {bound}(x,t)+\delta U(x,t)\).

Figure 14 shows the latitudinal distribution of \(T_{12_{ice}}(x,t)=T(x,t) + \delta T_{12_{ice}}(x,t),\) \(T_{23_{ice}}(x,t)=T(x,t)+\delta T_{23_{ice}}(x,t)\) and *T*(*x*, *t*), which is given by the equilibrium solution describing the current climate (Eq. (5)). \(T_{12_{ice}}\) and \(T_{23_{ice}}\) represent the equilibrium temperatures reached after the perturbations \(\delta T_{12_{ice}}\) and \(\delta T_{23_{ice}}\) are applied to the system. This latter achieves the climate state III, i.e. the condition of an ice-covered Earth with \(x_{s}=0\). However, for any *U*(*x*, *t*) smaller than \(U_\text {bound}(x,t)\) the climate system would remain in state I (current climate with \(x_{s0}=0.95\)).

Despite the differences between the PDE model and the model described in North et al. (1981), such as the overall climate sensitivity and the parametrization of the albedo [in North et al. (1981) a step function is employed for \(\alpha (x,x_s)\)], the results, reported in Fig. 15 of this paper and in Fig. 8 in North et al. (1981), are broadly comparable. In both cases the climate system shows two equilibrium states for current insolation conditions. As can be seen from Fig. 15, if the solar constant is decreased to 0.94, the unstable equilibrium state is reached and although the solar constant is increased again the ice line decreases further and the ice-covered state is reached. As it will be seen later, a much larger warming perturbation is required to lead the system back to the current climate state if an ice-covered state is reached. This outcome is in agreement with the literature where steady-state climate models are considered and suggests that only a \(6\%\) reduction of insolation is required to trigger the instability. Otherwise following the approach developed in this paper (Sect. 5), with the PDE model in Sect. 2.1, it is found that the overall reduction needs to be \(8.8\%\) to make the system unstable (Sect. 4). This discrepancy is due to the time-dependency of the PDE model employed Schneider and Gal-Chen (1973).

*Q*are broadly comparable with the normalized values shown in Fig. 15.

Finally, the recovery from an ice-covered state is investigated for completeness. The procedure described in Sect. 5 is applied again in order to estimate the variation of insolation necessary to drive the Earth’s climate from an ice-covered state (III) to the previous state (the current climate state I). Therefore, in this case, \(x_{s0}=0\), \(T_s=-20.44\,^{\circ } \text {C}\) and \(\lambda =-0.28\) [obtained through Eq. (25)] are employed. Again, a family of control functions which would trigger a recovery is found. In Fig. 16, several control functions are reported for values of \(C_1\) between 0 and 3. As before, the minimum control function can be found setting \(C_1=0\) and Fig. 17 is obtained.

Thus, considering the minimum control function reported in Fig. 17, it is estimated that an overall increase of \(U/Q_0=30\%\) is required to move the ice line from \(x_{s}=0\) back to \(x_s=0.95\). In particular, a maximum increase of \(30.1 \%\) is required at the equator and a minimum of \(10.6\%\) at the pole. This result can be also found in Caldeira and Kasting (1992).

*r*stands for recovery) provides the distribution of insolation increase required to move the climate system back to current conditions. In particular, \(U_r(x,t)\) is obtained considering Eq. (23) and applying the Lyapunov stability criterion as for the previous case. As before, the new equilibrium temperature can be computed through Eq. (35) and Fig. 18 is then obtained. In particular, the trend of the temperature of the ice-covered climate state is given by \(T_{23ice}\) whereas the new climate state reached is represented by \(T_{rec}\). It can be noted that the value of the equilibrium temperature at \(x=0.95\) is \(-10\,^{\circ } \text {C}\) and the overall equilibrium temperature of the new climate state is \( 24.1\,^{\circ } \text {C}\). In accordance with other results from the literature North et al. (1981), the new equilibrium state is found to be much warmer than the previous state with \(T\simeq 13\,^{\circ } \text {C}\) despite that the ice line is at \(x_{s0}=0.95\) in both cases. This is due to latitudinal diffusion of heat towards the poles and the strong ice-albedo feedback. Because of these phenomena, a considerable increase of solar radiation is required near the equator to move the ice line back to the pole and this causes a resulting warmer equilibrium climate state.

## 6 Conclusions

A time-dependent analytical model for the climate system with latitudinal resolution has been developed to assess closed-loop climate engineering strategies. The system investigated is a PDE model which can be analytically solved for any external forcing providing the latitudinal distribution of the temperature perturbation with time.

The model can be employed to investigate climate engineering strategies taking into account latitudinal disparities. High-fidelity numerical models for the climate can also be used to evaluate climate engineering strategies, but these models are computationally expensive. In contrast, the use of the PDE model provides a useful tool to rapidly asses SRM strategies, providing a clear understanding of the climate dynamics involved.

The PDE model developed in this paper is employed in three simulations [cases (1), (2), (3)] to explore multi-objective strategies with a PI feedback control. Several objectives were simultaneously minimized and the latitudinal response investigated considering a steady increase of \(\text {CO}_2\) concentration in the atmosphere (1pct\(\text {CO}_2\) scenario). The model provided analytical expressions for suitable control functions for three strategies, and proved to be effective when multi-objective analyses are considered. In fact, despite the simplicity of the model, results, which are broadly comparable with the literature, are found. The distributions of the control functions with latitude are obtained using the analytical solution of the PDE model and are again consistently comparable with the literature. In particular, the control responses and the temperature trends obtained with the analytical solution of the PDE model are comparable with those achieved through the CESM 1.0.2 (Community Earth System Model). This approach is considered as the verification of the general correctness and usefulness of the model developed.

Moreover, the model is used for the analysis of the zonal mean temperature. In agreement with the literature, it is found that the rms zonal mean temperature anomaly caused by the 1pct\(\text {CO}_2\) scenario decreases when more degrees of freedom are managed. Also, the PDE model can be employed to find the exact control law required and reduce the zonal mean temperature perturbation to zero. It is found that a non-uniform SRM distribution with increased control closer to the poles provides a more uniform offsetting of \(\text {CO}_2\)-induced warming and restores the mean temperature without over-cooling equatorial regions and under-cooling polar regions.

As a further example of the application of the PDE model, the upper limit on SRM is investigated through the analysis of the dynamics of the ice line. This analysis is of importance for climate engineering involving SRM because it shows that the extent of the insolation reduction commonly considered for SRM is far from the insolation reduction required to trigger instability of the climate. With respect to other similar calculations in the literature, where the insolation reduction required to achieve an ice-covered state is estimated, in this case, the whole family of control functions which would destabilise the system is found and analytical expressions are provided as a function of both latitude and time.

In this context, it is found that the minimum overall control effort required for an ice-covered state is approximately \(8.8\%\), which decreases towards zero as the climate cools. In particular, the maximum insolation reduction is needed at the equator (\(11.5\%\)) and the minimum at high latitudes (\(4.2 \%\)). The system then falls into a stable ice-covered state, where the global equilibrium temperature is estimated to be \(-32\,^{\circ } \text {C}\).

The use of the PDE model allows a clear and quick assessment of the boundaries of SRM, proving that the insolation reduction required to move the ice line to the equator is much higher than that considered for SRM deployment. It is therefore highly unlikely to accidentally force an ice-covered state.

## Notes

### Acknowledgements

Federica Bonetti acknowledges a University of Glasgow PhD scholarship and Colin McInnes acknowledges support from a Royal Society Wolfson Research Merit Award.

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