The Bern Simple Climate Model (BernSCM) is a free open-source
re-implementation of a reduced-form carbon cycle–climate model which
has been used widely in previous scientific work and IPCC assessments.
BernSCM represents the carbon cycle and climate system with a small
set of equations for the heat and carbon budget, the parametrization
of major nonlinearities, and the substitution of complex component
systems with impulse response functions (IRFs). The IRF approach
allows cost-efficient yet accurate substitution of detailed parent
models of climate system components with near-linear behavior.
Illustrative simulations of scenarios from previous multimodel
studies show that BernSCM is broadly representative of the range of
the climate–carbon cycle response simulated by more complex and
detailed models. Model code (in Fortran) was written from scratch
with transparency and extensibility in mind, and is provided open
source. BernSCM makes scientifically sound carbon cycle–climate
modeling available for many applications. Supporting up to decadal
time steps with high accuracy, it is suitable for studies with high
computational load and for coupling with integrated assessment
models (IAMs), for example. Further applications include climate risk assessment in
a business, public, or educational context and the estimation of

Simple climate models (SCMs) consist of a small number of equations,
which describe the climate system in a spatially and temporally highly
aggregated form. SCMs have been used since the pioneering days of
computational climate science to analyze the planetary heat balance

Recent applications of SCMs are often found in research in which computational resources are still limiting. Examples include probabilistic or optimization studies involving a large number of simulations, or the use of a climate component as part of a detailed interdisciplinary model. SCMs are also much easier to understand and handle than large climate models, which makes them useful as practical tools that can be used by non-climate experts for applications for which detailed spatiotemporal physical modeling is not essential. This applies to interdisciplinary research, educational applications, or the quantification of the impact of emission reductions on climate change.

BernSCM as a box-type model of the carbon cycle–climate system based
on impulse response functions. Heat and carbon taken up by the mixed ocean
surface layer and the land biosphere, respectively, is allocated to a series
of boxes with characteristic timescales for surface-to-deep ocean transport
(

An important application of SCMs is in integrated assessment models
(IAMs). IAMs are interdisciplinary models that couple a climate
component with an energy-economy model to simulate emissions and
their climate consequences. Another application of simple models

BernSCM is a zero-dimensional global carbon cycle–climate model built
around impulse-response representations of the ocean and land
compartments, as described previously in

BernSCM (Fig.

BernSCM simulates global mean surface temperature and the heat uptake
by the planet. The latter is equivalent to the net
top-of-the-atmosphere energy flux. Changes in the Earth's heat storage
in response to anthropogenic forcing are dominated by warming of the
surface ocean and the interior ocean

As with carbon, surface-to-deep transport is the rate-limiting step
for ocean heat uptake and thus for the adjustment of surface
temperature to radiative forcing. This transport is key to determine
the lag between realized warming and equilibrium warming

Non-

The present version 1.0 of BernSCM is fundamentally analogous to the
Bern model as used already in the IPCC Second Assessment Report,
Bern-SAR (whereas different versions of the Bern model family were
used in the more recent IPCC reports). BernSCM represents the relevant
processes more completely than Bern-SAR, thanks to additional
alternative representations of the land and ocean components, which
contain a more complete set of relevant sensitivities to temperature
and atmospheric

Here, BernSCM model simulations are compared to previous multimodel
studies. The model is run for an idealized atmospheric pulse

Together with this publication, BernSCM v1.0 is provided as an open-source Fortran code for free use. The code was also rewritten from scratch, with flexibility and transparency in mind. The model is comprehensively documented, and easily extensible. New alternative model components can be added using the existing ones as a template. A range of numerical solution schemes is implemented. Up to decadal time steps are supported with high accuracy, suitable for the coupling with emission models of coarse time resolution, for example. However, the published code is a ready-to-run stand-alone model, which may also be useful in its own right.

BernSCM offers a physically sound carbon cycle–climate representation, but it is small enough for use in IAMs and other computationally tasking applications. In particular, the support of long time steps is ideally suited to the application of BernSCM as an IAM component, as these complex models often use time steps on the order of 10 years.

BernSCM also offers a tool to realistically assess the climate impact
of carbon emissions or emission reductions and sinks, for example in
aviation, forestry

In this paper, we describe the model equations (Sect.

BernSCM simulates the relation among

The transport of carbon and heat to the deep ocean, as well as the
decay of land carbon, results from complex but linear to first order
behavior of the ocean and land compartments. These are represented in
BernSCM using IRFs (Green's
function). The IRF describes the evolution of a system variable after
an initial perturbation, e.g., the pulse-like addition of carbon to
a reservoir. It fully captures linear dynamics without representing
the underlying physical processes

The net primary production (NPP) of the land biosphere and the surface
ocean carbon uptake depend on atmospheric

The budget equation for atmospheric carbon is

The change in land carbon is given by the balance of NPP and decay of assimilated terrestrial carbon,

Carbon is taken up by the ocean through the air–sea interface
(

The net flux of carbon into the ocean is proportional to the gas
transfer velocity (

The global average perturbation in surface water

BernSCM simulates the deviation in global mean SAT from the
preindustrial state. SAT is approximated by the temperature
perturbation of the surface ocean

Climate sensitivity is an external parameter, as the model does not
represent the processes determining equilibrium climate response. RF
of

The response of a time-invariant linear system to a time-dependent
forcing

In BernSCM, an IRF is used to calculate the perturbation of heat and
carbon in the mixed surface ocean layer (mixed layer IRF;

Equation (

The IRFs above can be expressed as a sum of exponentials,

The ocean IRF contains a positive constant coefficient

IRFs of ocean (blue) and land (green) model components (without
temperature dependence). Ocean components are normalized to a common mixed-layer depth of 50

Inserting the formula (Eq.

The differential equation system (Eq.

The timescales of an IRF describing a linear system are equivalent to
the inverse eigenvalues of the model matrix of that system and may
also be interpreted in the context of the Laplace transformation

Thinking of IRF components as box models is conceptually
meaningful. The simple Bern 4-box biosphere model

The IRF representation is, strictly speaking, only valid if the
described subsystem is linear and the timescales of the system are
time invariant. Then, the response function

The interpretation of the IRF representation as a box model provides
a starting point for considering nonlinearities in the response. To
account for nonlinearities, the response timescales

Varying coefficients have been successfully implemented and tested for
the HRBM land component and its decay IRF

The carbon cycle–climate uncertainty of simulations with BernSCM can
be assessed in two ways. First, to assess structural uncertainty,
different substitute models for the ocean and land components can be
used. Currently, this approach is quite limited by the set of
available substitute models (see Appendix

The uncertainties of the global carbon cycle concern the sensitivity
of the modeled fluxes of carbon and heat to changing atmospheric

As for the ocean, the uncertainty of heat uptake into the surface
ocean is treated in terms of climate sensitivity
(Eq.

Fraction of realized warming (temperature divided by the equilibrium
temperature for the current RF) for idealized experiments with prescribed
atmospheric

Similar to previous studies using models from the Bern family

All temperature and

All sensitivities are set to zero (except for the ocean

Only

Only temperature dependencies are considered in the land module (NPP, decay).

The climate response of BernSCM is illustrated using idealized
simulations with prescribed forcing. One series of simulations (a) was
run for

IRFMIP pulse response range compared to BernSCM range for parameter
uncertainty (colors according to legend) and structural uncertainty, with
model versions HILDA–HRBM (solid lines), HILDA–4-box (dots), and Princeton–HRBM
(dashed). Standard climate sensitivity is 3

In BernSCM, the fraction of realized warming depends primarily on the
choice of climate sensitivity and is qualitatively similar for the
different model setups. Such a clear relationship is not seen in the
EMS and EMICs. Thus the structural uncertainty and model differences
of complex models are not fully represented in BernSCM. The BernSCM
climate response to abrupt warming (Fig.

Coupled carbon cycle–climate models can be characterized and compared
based on their response to a

The IRFMIP pulse experiment was repeated with BernSCM, exploring
parameter uncertainty of the carbon cycle (Sect.

The AF simulated with BernSCM broadly agrees with the set of
simulations from IRFMIP. At 100 years after the pulse, the AF is 0.40
(0.34–0.57) for a climate sensitivity of 3

The BernSCM SAT response also broadly agrees with IRFMIP. The standard
coupled simulation is somewhat lower than the IRFMIP median, which is
explained in part by the climate sensitivity (3

Land, ocean, and airborne fractions of the 100 Gt C

Figure

Climate models with explicit and detailed carbon cycle components
exhibit a wide range of responses, as shown in the intercomparison
studies of climate models with a detailed carbon cycle, C4MIP

The climate–carbon cycle feedback is measured by the feedback metric

The C4MIP study used a SRES A2 emission scenario to compare the carbon
cycle sensitivities of a range of models. As in the C4MIP exercise,
BernSCM was run for SRES A2 without any non-

BernSCM simulations of the SRES A2 scenario used for C4MIP, with
a climate sensitivity of 2.5

The BernSCM sensitivity setups can be expressed in terms of the C4MIP
sensitivity metrics: T-only corresponds to

The land carbon uptake until 2100, under the different BernSCM
configurations, varies over 500 Gt C (Fig.

We simulated illustrative scenarios from two recent multimodel
studies, C4MIP and IRFMIP, to compare BernSCM to the literature of
carbon cycle–climate models. The results show that BernSCM is broadly
representative of the current understanding of the global carbon
cycle–climate response to anthropogenic forcing (in a time-averaged
sense that does not address internal variability). The BernSCM
uncertainty range in

As Fig.

BernSCM does not explicitly distinguish between surface atmosphere and
surface ocean temperature to compute global mean SAT perturbation. This is in contrast to some energy balance
calculations used to analyze results from state-of-the-art ESMs

C4MIP sensitivity metrics. The BernSCM range covers the carbon cycle
settings as discussed in Sect.

Currently, a limited set of substitute models is available and
included with BernSCM. The simple structure and open-source policy of
BernSCM allows users to address these current limitations according to
the needs of their applications. More components can be added using
the existing ones as a template. This requires the specification of
the IRF and the parametrization of gas exchange for the surface ocean
or NPP for the land biosphere

Ocean transport is known to vary under climate change with some
consequences for heat and carbon uptake

A distribution of timescales applies to ocean transport processes as
evidenced by observations of transient and time-dependent tracers such
as chlorofluorocarbons and bomb-produced and natural radiocarbon and
biogeochemical tracers

The BernSCM model may be extended to model perturbation in the
signatures and exchange fluxes of the carbon isotopes

A potential future application of BernSCM is to use it as an emulator
of the global long-term response of complex climate–carbon cycle
models by adding the corresponding substitute model components.
Additionally, pattern scaling can be applied to transfer the global
mean temperature signal into spatially resolved changes in surface
temperature, precipitation, cloud cover, etc., exploiting the
correlation of global SAT with regional and local changes

The addition of further alternative model components will extend the
structural uncertainty that can be represented with
BernSCM. A sufficient coverage of structural uncertainty could allow
the interpolation among alternative model components to represent
uncertainty with scalable parameters (and removing the distinction
between structural and parameter uncertainty). Such a parametrization
of the uncertainty would enhance the possibilities for probabilistic
applications of BernSCM, although more sophisticated models are
available for observation-constrained probabilistic quantification of
climate targets

BernSCM is a reduced-form carbon cycle–climate model that captures the
characteristics of the natural carbon cycle and the climate system
essential for simulating the global long-term response to
anthropogenic forcing. Simulated atmospheric

Due to its structural simplicity and computational efficiency, BernSCM has many potential applications. In combination with pattern scaling, BernSCM can be used to project spatial fields of impact-relevant variables for applications such as climate change impact assessment, coupling with spatially explicit land biosphere models, etc. With alternative numerical solutions of varying complexity and stability to choose from, applications range from educational to computationally intensive integrated assessment modeling. BernSCM also offers a model-based alternative to global warming potentials for estimation of the climate impact of emissions and can be used to quantify climate benefits of mitigation options by applying emissions- or concentration-driven simulations.

The generic implementation of linear IRF components offers a transparent, extensible climate model framework. Current limitations concern the number of available substitute models (limiting the uncertainty range represented), and ocean transport not influenced by climate change. An addition of further alternative model components and more flexible representation of sensitivities in terms of continuously variable parameters would further increase the models' usefulness, for example for probabilistic applications.

The source code of the Bern Simple Climate Model is available from the GitHub repository at

Currently available ocean components include substitute models for
the high-latitude exchange/interior diffusion–advection model

Model variables.

Model parameters.

Mixed-layer IRF/Box parameters.

Land C stock IRF/Box parameters.

Ocean surface

Ocean surface

Currently available land biosphere components include substitute
models for the High-Resolution Biosphere Model (HRBM)

For the HRBM model, temperature-dependent IRF/box model parameters
as given by

Net primary production for HRBM is given by

Net primary production for the 4Box model is described after

For the solution of the pulse-response
equation (Eq.

First,

Second, for longer time steps, a better approximation is obtained by
assuming linear variation in

For the solution of the BernSCM model equations, both explicit and implicit time stepping is implemented.

The stability requirement for the numerical solution depends on
the equilibration time for the ocean surface

For the tested scenario range, the explicit solution is stable at a time step on the order of 0.1 year, for which the piecewise constant approximation is accurate. For a larger step size, an implicit solution is required to guarantee stability.

The piecewise constant approximation is adequate for time steps up
to 1 year, and the piecewise linear approximation is adequate for up to decadal
time steps. An overview of the performance of three representative
settings (set at compile time) for the C4MIP A2 scenario is given
in Table

Performance and accuracy for time steps of 1–10 years relative to
a reference with a time step of 0.1 year. The reference simulation is solved
explicitly; otherwise an implicit solution was used. The average execution
time of the time integration loop is given as a fraction of the explicit
case. For atmospheric

The explicit solution is only implemented for the piecewise
constant approximation (Eq.

First consider the equation system for carbon, assuming temperature
to be known (or neglecting temperature dependence of model
coefficients). Equation (

To solve the implicit system, the nonlinear parametrizations need
to be linearized around

The system is completed with the discretized budget
equation (Eq.

After calculating the “committed” values

The order of these equations matters, as the updated variables are successively inserted into the following equations. The land part is solved first, and can be substituted by an explicit step or a separate model, while keeping the ocean step implicit.

An implicit time step is also implemented for calculating SAT from
RF (again, solving RF from SAT is also implemented but not
discussed here). RF

The case of piecewise linear
approximation (Eq.

BernSCM allows for temperature-dependent model parameters for IRF-based substitute models. This generalization of the IRF approach
is possible using a box model form (Sect.

BernSCM updates any temperature-dependent model parameters by approximating the current temperature

The authors declare that they have no conflict of interest.

This work received support from the Swiss National Science Foundation (no. 200020_172476). Edited by: Carlos Sierra Reviewed by: Holger Metzler and one anonymous referee