Effective Climate Sensitivity Distributions from a 1D Model of Global Ocean and Land Temperature Trends, 1970-2021

A 1D time-dependent forcing-feedback model of temperature departures from energy equilibrium is used to match measured ranges of global-average surface and sub-surface land and ocean temperature trends during 1970–2021. In response to two different radiative forcing scenarios, a wide range of three model free parameters are swept to produce �ts to a range of observed surface temperature trends from four different land datasets and three ocean datasets, as well as deep-ocean temperature trends and borehole-based trend retrievals over land. Land-derived effective climate sensitivities (EffCS) are larger than those over the ocean, and EffCS is lower using the newer Shared Socioeconomic Pathways (SSP245, 1.94 deg. C global EffCS) than the older Representative Concentration Pathway forcing (RCP6, 2.60 deg. C global average EffCS). Diagnosed EffCS increases with increasing ocean or land heat storage, with close to 1 deg. C EffCS increase associated with ocean heat storage, but only 0.13 deg. C increase from land heat storage. The strongest dependence of the EffCS results is on the assumed radiative forcing dataset, underscoring the role of radiative forcing uncertainty in determining the sensitivity of the climate system to increasing greenhouse gas concentrations from observations alone.


Introduction
The determination of the sensitivity of the climate system to increasing greenhouse gas concentrations, usually stated in terms of the surface temperature change in response to a doubling of pre-industrial levels of atmospheric CO 2 (2XCO2), has remained elusive.In the forcing-feedback paradigm of climate change departures from global energy balance, the top-of-atmosphere (TOA) radiative energy imbalance N is the sum of an imposed radiative forcing F and a feedback response -λΔT, N = F -λΔT, (1) where the net feedback factor λ determines the climate sensitivity and ΔT is the global average surface temperature departure from the normal equilibrium state (Charney et al. 1979).For example, the radiative forcing from a doubling of atmospheric CO 2 is generally accepted to be 3.7 W m − 2 (Forster et al. 2021) and as the system warms over many centuries the TOA energy imbalance N is removed, and a nal equilibrium climate sensitivity change in temperature ΔT is achieved at F/λ.For over 30 years the range of equilibrium climate sensitivities (ECS) diagnosed either from theory (3D Earth System Models, ESMs) or from observations has persisted over a broad range between 1.5 and 4.5 deg.C, with a few outlier estimates (Meehl et  The EffCS uncertainty on a theoretical level arises from the complexity of the feedback responses of the climate system to a radiative imbalance, such as how clouds change to either amplify or dampen warming.On an observational basis, EffCS uncertainty comes from a lack of accurate knowledge of both the radiative forcing and the temperature response of the system to that forcing over the last 50 to 100 years or more.Gregory et al. (2020) addressed reasons why such estimates can produce biased results, for example due to the in uence of major volcanic eruptions.
Alternatively, one can instead examine shorter-term interannual co-variations in TOA radiative ux and temperature to estimate λ, but non-feedback variations in TOA radiative ux de-correlate those variations leading to underestimates of λ made through standard least-squares linear regression (e.g.Spencer and Braswell 2011).
Further complicating observational diagnosis of sensitivity is the large heat capacity of the ocean, causing a delay in surface warming compared to if there was no sub-surface energy storage, which necessitates accurate measurements of ocean heat content over most of the global oceans' volume.
While heat storage by the landmass is usually ignored in such evaluations, here we include an estimate of its impact on diagnosed climate sensitivity.
Finally, uncertainty in diagnosing ECS from observations arises from multi-decadal time scale internal uctuations in the climate system which can cause 10-20 year periods with either strong warming or no warming unrelated to the system's long-term response to anthropogenic forcing (e.g.Meehl et al. 2013).
The complexity of the wide range of processes which determine climate sensitivity, combined with the rather wide range of sensitivities exhibited by climate models, leads to a need for simple alternative methods for examining what range of sensitivities are implied by the observed rates of global warming.
Here we use a 1D time-dependent model of temperature departures from energy equilibrium, over land and ocean separately, to diagnose EffCS for a range of observed temperature trends over land and ocean, utilizing two signi cantly different radiative forcing datasets.The model could be considered the simplest The energy budget approach to estimating EffCS is similar to that of Lewis and Curry (2018, hereafter LC18), which obtained EffCS values ranging from 1.5 to 1.8 deg.C by examining 100 + year time scale changes in temperature and assumed forcing.In contrast to LC18, we use a time dependent model, which allows us to examine features such as the acceleration of deep-ocean (0-2000m) warming in recent decades (Cheng et al. 2019).The other difference is that we focus on the most recent 50 years (1970-2021) during which radiative forcing from greenhouse gas increases has been the largest and when observed deep-ocean temperature changes have the least measurement error.This hopefully maximizes the signal-to-noise of the ECS estimation, keeping in mind that the time period cannot be too short otherwise natural interannual climate variations can corrupt the diagnoses.While the largest volcanic eruption in modern history occurred during this period (Pinatubo in 1991), it was positioned near the middle of the period, hopefully reducing its impact on the computed temperature trends and associated uncertainties in volcanic radiative forcing (Gregory et al. 2020).
While the simplicity of the model allows simulations to be carried out quickly, it is at the expense of not knowing what speci c feedback processes determine EffCS.Only their net effect on the TOA radiative ux and temperature are determined.To include the effect of uncertainty in the history of radiative forcing over that period (which is quite large, due mostly to sulfate aerosol forcing uncertainty, Smith and Forster 2021) two substantially different radiative forcing histories are included.Thousands of simulations are carried out spanning the full range of observed temperature trends and potential range of model free parameters to produce frequency distributions of diagnosed EffCS for all model ts to the observational data.

The 1d Model And Radiative Forcing Scenarios
The 1D time-dependent model roughly follows that of Spencer and Braswell (2014), but with a simpli ed vertical heat transfer scheme.It computes monthly temperature departures from assumed energy equilibrium beginning in 1765 in three layers over land and ocean separately (shown schematically Fig. 1), with adjustable heat transfer coe cients between layers which act to reduce temperature departure gradients between layers.While the model could allow for land-ocean energy exchanges to be included, this will not be explored here due to a lack of accurate knowledge of changes in energy ows between land and ocean during warming.It should be kept in mind that if there is a change in the ows of energy between land and ocean through atmospheric transport, this would impact the diagnosed values of EffCS over land and ocean separately, but should have little impact on the global average EffCS.The assumed layer depths over land and ocean are also shown in Fig. 1.
The layer thicknesses represent a con guration which captures temperature changes on interannual and longer time scales.The model equation for the rst layer temperature (T 1 ) departures from equilibrium (dΔT 1 /dt), for either land or ocean, is where C p1 is the bulk heat capacity of the rst layer (land or ocean); F represents time-dependent radiative forcings; λ is the net feedback parameter (Forster and Taylor 2006;Forster and Gregory 2006) which is allowed to be different over land and ocean; and, h 1 is the effective vertical heat transfer coe cient (W m − 2 K − 1 ) between the rst two model layers.Note that the heat transfer coe cients have the same units as the net feedback parameter (W m − 2 K − 1 ).
The second and third model layer temperatures are governed respectively by effective radiative forcing values come from the tables in Annex III of IPCC (2021b).As seen in Fig. 2, these represent quite different radiative forcing scenarios, especially in terms of the rate of growth during the period we will be addressing temperature trends .To obtain a yearly time series from the SSP245 best estimates of effective radiative forcing (ERF, dots in Fig. 2) piecewise-linear time-dependent adjustments to the yearly-resolution RCP6 radiative forcing history are used to produce a yearly resolution SSP245 that includes the major volcanic eruptions seen in RCP6, and approximately matches the benchmark years in which the SSP245 values were tabulated.The observed temperature trends during 1970-2021 are on the low side of those produced by 36 CMIP6 models (Fig. 3), especially for the ocean.The land/ocean warming ratio (WR) based upon these trends averages 1.45 across all of the models, compared to 2.02 from the center of the observational dataset ranges.Knowledge of sub-surface warming over land is not as well established as it is for the ocean.To provide a target for model sub-surface warming, the Northern Hemisphere mid-latitude borehole study of Harris and Chapman (2001) was used.In that work, a retrieved pro le of sub-surface temperature trends showed warming extending as deep as 200 m, reproduced in Fig. 4.

Validation Datasets
Their retrieved pro le of warming showed that the surface (and thus our rst layer of depth 2m of soil) has warmed at about 2.56 times the rate of the uppermost 100m, while the ratio of surface warming to 100-200 m layer warming is about 30.The land effective heat transfer coe cients in Eqs.2-4 are adjusted in the model simulations to target these ratios, +/-20% and +/-30% respectively, thus producing a 1D land model temperature trend pro le shape with depth that would approximately match the retrieved pro le shape in Fig. 4.

1d Model Experiments
The model is initialized in 1765, and each of the model's three free parameters (λ and two inter-layer heat transfer coe cients) was swept independently with 70 closely-spaced values covering a range su cient to encompass all 1D model temperature trend matches with observations (1970-2021).The resulting frequency distribution of diagnosed EffCS values from the SSP245 effective radiative forcing history is shown in Fig. 5a for land and Fig. 5b for ocean, while those forced with the RCP6 radiative forcing history are shown in Fig. 5c for land and Fig. 5d for 2017) dataset (Fig. 6) suggests more acceleration of ocean heat storage since approximately 1990 with the SSP245 effective radiative forcing than the RCP6 radiative forcing, even though both have the same trends over the 1970-2021 period.
The 1D model results here produce higher EffCS values than those reported by LC18, which ranged from 1.5 to 1.8 deg.C for the global average compared to 1.94 deg.C here using the SSP245 effective radiative forcing estimates.This difference is not surprising given the difference in time periods addressed, and our inclusion of deep-ocean (below 2000 m) and deep-land heat storage, both of which will act to increase the estimate of EffCS.Also, the strong dependence of EffCS on which radiative forcing history is used is likely to also affect the results.
Finally, of some interest is the effect that ocean (or land) heat storage has on diagnosed EffCS.When the model is run with no heat storage below the rst layer (50 m deep for ocean 2 m deep for land), the diagnosed EffCS is reduced by 0.13 deg.C over land and by 0.90 deg.C over ocean.Thus, land heat storage has little impact on EffCS, but considerable impact on ocean (and thus global average) EffCS.

Summary And Discussion
A 1D time-dependent model of global monthly average temperature departures from energy equilibrium was forcing with the newer SSP245 and older RCP6 radiative forcing scenarios.The model adjustable parameters are the net radiative feedback parameter (which determines the effective climate sensitivity) and two heat transfer coe cients which determine the rate of heat transfer between the three model layers, land or ocean.When the model is run with a large combination of potential values of the three free parameters, the model-produced temperature trends during 1970-2021 match a variety of observed trends (with error bounds), producing frequency distributions of EffCS over land and ocean, separately.Diagnosed EffCS is higher over land than over ocean, consistent with greater observed warming trends there.Global average EffCS is 1.94 deg.C. for the SSP245 ERF scenario, and 2.60 deg.C for the RCP6 RF scenario.These are near the lower end of the most recent climate sensitivity estimates from IPCC (2021a) of 2.5 to 4.0 deg.C, although the possibility of EffCS increasing in the future cannot be addressed from the historical data analyzed here.The 1D model results produce higher EffCS values than those reported by LC18, which ranged from 1.5 to 1.8 deg.C, despite some similarities in methodology.This difference could be from our inclusion of land heat storage to match borehole measurements and ocean heat storage below 2,000 m depth, both of which will increase EffCS diagnoses.Also, we use temperature trends over the recent period (1970-2021) instead of differences over a 100 + year time scale to focus on a time period with the greatest radiative forcing change and the most accurate observational data.The results above suggest large uncertainty in diagnosing EffCS from historical land and ocean temperature data due to primarily to uncertainty in real-world radiative forcing.Given that ERF from increasing CO 2 is well established, the uncertainty is likely to be in the aerosol radiative forcing over the last 50 + years.
EffCS diagnosed over land is uniformly higher than that over the ocean, which is consistent with longterm integrations from ESMs that show, irrespective of radiative forcing scenario, ~ 50% greater warming over land than ocean, for a land-ocean warming ratio (WR) of approximately 1.5 (Wallace and Joshi 2018 and references therein).For the time period addressed here (1970-2021) the land rate of warming is approximately twice that of the ocean, for a WR of 2.0.But the ratio of the 1D model-diagnosed EffCS is much smaller, 1.11 for the SSP245 forcing scenario and 1.30 for the RCP6 radiative forcing scenario.
The effect of heat storage on diagnosed EffCS is much larger over the ocean than over land, by a factor of approximately seven, with deep-land heat storage increasing EffCS by 0. Ethics approval/declarations: Not applicable.Consent to participate.Not applicable (?) (I don't know what this refers to, as there is no reference to "participate" or "participation" anywhere in the "Submission guidelines" at https://www.springer.com/journal/704/submission-guidelinesConsent for publication: The authors grant TAAC sole rights to publish the submitted manuscript and we agree to the terms outlined under "Ethical Responsibilities of Authors".(I am not sure what this refers to in the "Submission guidelines", either).
Data Availability: The 1D model, including the input datasets, input parameters, and output data, will be available at http://www.nsstc.uah.edu/data/roy.spencer al. 2020, and references therein).The most recent estimates from ESMs participating in the sixth Coupled Model Intercomparison Project (CMIP6, Eyring et al. 2016) cover the widest range yet (1.8 to 5.7 deg.C) although the CMIP6 expert evaluation of the most likely range has narrowed to 2.5 to 4.0 deg.C, with a best estimate of 3.0 deg.C (IPCC 2021a).Due to the long time scale (centuries) required for the deep ocean to reach a new equilibrium state, the possibility that feedbacks can change on multi-century time scales, and the differences in e cacy of various forcing agents (e.g.aerosols vs. CO 2 ), a shorter-term "effective" climate sensitivity (EffCS, e.g.Gregory et al. 2020) is usually preferred over ECS as a more practical measure for energy policy decisions and mitigation planning.
approximation of ESMs where time-dependent equations are used to compute temperature tendencies in response to sources and sinks of energy.The simplicity of the model allows rapid computation of the sensitivity of EffCS to choices of assumed TOA radiative forcing and temperature datasets.Here we include deep-ocean (below 2,000 m) storage of heat as well as deep-land storage to 200 m depth, based upon borehole temperature measurements.The storage of heat in the global landmass is still not well handled by ESMs, which contain Land Surface Models (LSMs) with bottom boundary condition placement (BBCP) at only 2 to 10 m in depth (Cuesta-Valero et al. 2016; Burke et al. 2020), despite borehole evidence of warming to 200 m depth over recent centuries (National Research Council 2006; Harris and Chapman 2001).

where h 2
is the heat transfer coe cient between the second and third model layer.The Cp values are the product of the ocean or land volumetric heat capacity (4.186 MJ m − 3 K − 1 for seawater, and 2.4 MJ m − 3 K − 1 over land (MacDougall et al. 2010; Legutke and Voss 1999) and the thickness of the layer.Two time-varying histories of radiative forcing are used, one from CMIP5 and one from CMIP6.The CMIP5 forcing is the Representative Concentration Pathways (RCP) 6.0 scenario, for which Meinshausen et al. 2011 produced the yearly time resolution estimates, here interpolated to monthly.The SSP245 The ocean surface temperature trends the 1D model simulations are matched to come from three datasets: HadCRUT5 (Morice et al. 2020), NOAA Global Temp (Menne et al. 2018); and Berkeley (Rohde and Hausfather 2020).For land surface air temperature four datasets were used: the three just mentioned, and GISTEMP v4 (GISTEMP Team 2022).To qualify as a match to observations, the ocean model's rst layer temperature trend needs to match any trend within the range represented by the four observational land surface temperature datasets (+ 0.253 to + 0.306 deg.C per decade in NOAA Global Temp and GISS Temp, respectively), while the sea surface temperature trends must fall within the range of the three SST datasets (+ 0.114 to + 0.162 deg.C per decade in NOAA Global Temp and Berkeley, respectively).
The ocean 0-2000 m temperature trends of Cheng et al. (2017) must be matched by the model to within +/-20% which covers the full range of trends in the datasets of Ishii et al. (2017), Dominigues et al. (2008) and Levitus et al. (2012).The bottom ocean layer in the model (2000 m to ocean bottom) is required to warm a total of 0.012 (+/-30%) deg.C from 1990 to 2021 to agree with the Cheng et al. (2017, based upon Purkey and Johnson 2010 and von Schuckmann et al. 2020) estimate of 30 ZJ of warming below 2,000 m depth since 1990.
ocean.Note that the newer SSP245 effective radiative forcing estimates produce considerably lower EffCS values than do the RCP6 estimates, with a central estimate of 2.08 deg.C (range 1.88-2.34)for land and 1.88 deg.C (range 1.33-2.89)for ocean, for a global average of 1.94 deg.C. The RCP6 radiative forcing scenario produces 3.10 deg.C (range 2.82-3.46)for land, 2.39 deg.C (range 1.80-3.36)for ocean, with a global average EffCS of 2.60 deg.C. Note these ranges of diagnosed EffCS are rather large, a re ection of the uncertainties in the variety of observational dataset temperature trends over land and ocean.The EffCS diagnosed from SSP245 here (1.94 deg.C) is qualitatively consistent with the somewhat lower observed trends in Fig.3compared to the CMIP6 models, keeping in mind that deep-ocean (and land) heat storage differences with the CMIP6 models will also affect the comparison.Only 2 of the 36 CMIP6 models in Fig.3have ECS lower than 2.0 deg.C, INM-CM4-8 and INM-CM5-0 (Meehl et al. 2020) The ts of the 1D model 0-2000m ocean layer to the Cheng et al. ( 13 deg.C and deep-ocean heat storage increasing EffCS by 0.98 deg.C. The 1D model presented here offers a simple yet energetically consistent method for establishing what range of effective climate sensitivities is represented by the variety of observational datasets of temperature trends over land and ocean, at the surface and in the sub-surface.It can provide a baseline for comparison to ESMs and permit evaluation of how dependent climate sensitivity estimates are to both observational datasets and to assumed radiative forcing histories.Unfortunately, for the reasons stated above, these observational energy-based estimates of EffCS cover a rather wide range, consistent with the conclusions of Gregory et al. (2020) that a longer period of time (e.g.into the 2030s) without a major volcanic eruption might be required to reduce the uncertainty in EffCS diagnoses based upon observations.Declarations Funding: This research was funded though U.S. Department of Energy contract DE-SC0019296 and through the Alabama O ce of the State Climatologist.Competing Interests: The authors have no relevant nancial or non-nancial interests to disclose.

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Figure 3 Surface
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