Radiative kernels describe how a small change in an atmospheric state variable affects the Earth's energy balance . They allow an analysis of climate feedbacks or forcing adjustments from standardised climate model diagnostics such as those from the Coupled Model Intercomparison Projects (CMIPs). The use of radiative kernels is efficient, removing the need for time- and memory-consuming calculations of climate feedbacks online through partial radiative perturbation calculations or offline using a stand-alone version of the model radiative transfer code .

A radiative kernel KX is in effect a four-dimensional (time, height, latitude, longitude) array describing how radiation fluxes R change with an atmospheric state variable X: 1KX(t,z,y,x)=∂R∂X(t,z,y,x).

Although strictly not a partial differential equation, this statement provides a concise written form and is used by others e.g.. R may be upwelling, downwelling or net, shortwave or longwave radiation changes at any atmospheric level. Most commonly net top-of-atmosphere (TOA), surface and tropopause-level fluxes are of greatest interest. X here represents atmospheric temperature (Ta), surface (skin) temperature (Ts), water vapour (q) and surface albedo (α). Although other (non-cloud) variables may also be relevant, the majority of adjustments are expected to be captured under this framework . For determining adjustments to a radiative forcing AX, the kernel KX is multiplied by the change in atmospheric state variable ΔX between two integrations of a climate model such that 2AX=KXΔX. ΔX is calculated as the difference of two atmosphere-only climate model integrations using climatological sea-surface temperatures and sea-ice distributions, one of which is driven by a forcing perturbation (e.g. a quadrupling of CO2) and the other a control. For temperature and albedo the adjustment is linear with ΔX and logarithmic for water vapour (, and Supplement of , describe how the adjustment to water vapour is applied in practice). For determining climate feedbacks λX, the perturbation is normalised by the change in global mean near-surface air temperature T such that 3λX=KX∂X∂T. The individual contributions from each feedback component λX contribute the total climate feedback λ=λTa+λTs+λq+λα+λc where “c” represents cloud feedback in the forcing-feedback representation of the Earth's energy budget ΔN=F-λΔT. Here, ΔN is the Earth's energy imbalance and F the effective radiative forcing. Likewise, the effective radiative forcing can be decomposed into 4F=Fi+ATa+ATs+Aq+Aα+Ac, with Fi being the instantaneous radiative forcing (IRF).

Usage of radiative kernels assumes that radiative perturbations change linearly with changes in atmospheric state. Where perturbations are small, linearity is an appropriate assumption both for feedbacks and adjustments .

Cloud adjustments and feedbacks cannot be determined directly using atmospheric state kernels. They may be diagnosed using the cloud kernel based on International Satellite Cloud Climatology Project (ISCCP) simulator diagnostics or from the residual of all-sky and clear-sky radiative kernels . For adjustments this calculation is 5Ac=(F-Fclr)-(Fi-Ficlr)-∑X∈{Ta,Ts,q,α}(AX-AXclr), where the “clr” superscript represents fluxes calculated in the absence of clouds. In Eq. (), the instantaneous radiative forcing must be known or estimated. This method is commonly used, requiring the production of all-sky and clear-sky kernel sets to calculate all-sky and clear-sky adjustments.

This paper introduces the top-of-atmosphere and surface radiative kernels from the HadGEM3-GA7.1 model. HadGEM3-GA7.1 has 85 vertical levels up to 85 km vertical height (about 0.005 hPa). The construction of the HadGEM3-GA7.1 kernel was motivated by the observation that adjustments to a doubling of CO2 in models participating in the Precipitation Driver and Response Model Intercomparison Project (PDRMIP; ) were around 0.3 W m-2 larger using the ECMWF-Oslo kernel than other kernels used in the same study Supplement Fig. 3. The ECMWF-Oslo kernel was built from ECMWF-Interim reanalysis data which have 60 vertical levels up to 0.1 hPa. Most other kernels used in were derived from climate models with a lower model top, and it is possible that low-top kernels were underestimating the stratospheric adjustment in .

For stratospheric temperature adjustments we compare the HadGEM3-GA7.1 kernel to other kernels in the literature using available 4×CO2 results from climate models contributing to the Radiative Forcing Model Intercomparison Project (RFMIP). We find that, in general, kernels based upon climate models with a high stratospheric resolution are better at resolving the stratospheric adjustment to a CO2 forcing.

One year of a pre-industrial, atmosphere-only (i.e. with climatological sea-surface temperatures and sea-ice distributions) integration of the HadGEM3-GA7.1 general circulation model was run. HadGEM3-GA7.1 is the atmospheric component of the HadGEM3-GC3.1 physical model and UKESM1.0 Earth system model that represents the UK research community's contribution to CMIP6. The model was run at LL (N96) resolution with a latitude–longitude grid of 1.25∘ by 1.875∘ and 85 vertical levels extending up to 85 km (approximately 0.005 hPa) and a native model time step of 20 min. A pre-industrial base climatology for the kernels was chosen because the first identified use case for this kernel set was the RFMIP-ERF Tier 1 single-forcing experiments . These experiments compare positive (e.g. greenhouse gas) and negative (e.g. aerosol) forcing perturbations with respect to a pre-industrial control baseline.

Model diagnostics of air temperature, specific humidity, surface (skin) temperature, surface albedo (ratio of broadband upwelling to downwelling shortwave surface radiation), model level pressure, surface pressure, cloud fraction, cloud water content, cloud ice content, effective (time-averaged) solar zenith angle and grid box daylight fraction every two model hours were saved. Two-hourly sampling is considered to give an appropriate representation of the diurnal cycle of clouds and reduce biases due to variations in solar geometry with longitude compared to longer time steps, while keeping computational demands to a minimum. These model outputs were transplanted into an offline version of the SOCRATES radiative transfer code (version 17.03; ) and top-of-atmosphere and surface radiative fluxes calculated for each 2 h time step in both the shortwave and longwave spectra, for all sky and clear sky. SOCRATES is a broadband radiation code that uses six bands in the shortwave and nine bands in the longwave and is the same radiation scheme used in the HadGEM3-GC3.1 and UKESM1.0 climate models. Aerosols were neglected and greenhouse gases, including the prescribed CMIP6 monthly climatology in ozone concentrations, were set to their pre-industrial (1850) values. Following the protocol for RFMIP , sea-surface temperatures and sea-ice distributions from 50 years of the HadGEM3-GC3.1 coupled model were used to build the climatology . The kernel is therefore dependent on two aspects of the HadGEM3-GA7.1 model: the pre-industrial background climatology (including clouds) and the broadband version of the radiation code.

To build the kernel, each vertical level of the model on each 2 h time step was perturbed separately, firstly by 1 K for air temperature and secondly by a perturbation in specific humidity that maintains relative humidity for an increase in 1 K (without actually changing the layer temperature). The surface temperature and surface albedo were also perturbed by 1 K and 1 % (additive) individually each time step. For each perturbation, surface and TOA fluxes are again saved for clear sky and all sky in the shortwave (SW) and longwave (LW), and the difference compared to the control simulation gives the radiative kernel for each model level or surface. Building the kernels took in total approximately 3 months of computing time on 24 processors on the University of Leeds “cluj” Linux cluster. Running the base HadGEM3 model at higher (MM) resolution, with 5 times as many grid points as LL, was not determined to be necessary, as kernels are usually ultimately regridded to the resolutions of other CMIP6 models which are approximately as coarse as LL (e.g. Table 1 in ).

Following this, the air temperature and water vapour kernel outputs were normalised by multiplying by 10 000 Pa / pthick, where pthick [Pa] is the thickness of each level in pressure coordinates. This allows the 85-level native model kernel to be reduced down to the 19-level standard CMIP6 pressure levels by providing a weighted average contribution to each pressure level. The kernels are further averaged by month. In the 19-level format they can be used with standard “Amon” model output from any CMIP6 model, which is one of the key advantages of radiative kernels.

Figure  shows the air temperature kernel for the stratosphere and upper troposphere for a selection of kernels available in the literature (Fig. ). In all cases, radiative kernels have been interpolated from their native vertical resolution (except for CCSM4, which is available only on the standard 17 CMIP5 pressure levels) to the 19 CMIP6 pressure levels for consistency with CMIP6 model output. For our calculations of stratospheric temperature adjustment, where kernels do not extend up to the 1 hPa top level of CMIP6 model output, kernels have been extended upwards using the value from the highest level where data do exist, but in Fig.  missing data have been masked out. This extending upwards of the top level has been applied previously in adjustment calculations where the top level of the climate model is higher than the top level of the kernel (e.g. in ). However, extending the top level of a radiative kernel upwards cannot make up for the fact that more radiation is emitted to space from the upper stratosphere for each additional K of temperature change. For kernels built from underlying atmospheric profiles where the top of the profile is not sufficiently high or with too coarse a resolution in the stratosphere, this additional upper stratospheric cooling is missed. In Fig. , it can be seen that the kernels based on a high-top atmospheric model with a high number of native model levels – ECHAM6, ECMWF-Oslo, ECMWF-RRTM and HadGEM3-GA7.1 – have a marked increase in both the magnitude and the rate of negative LW outgoing flux at the 5 and 1 hPa levels.

The consequences for a greenhouse-gas-induced stratospheric cooling are such that the additional stratospheric adjustment from greater cooling high in the stratosphere is not accounted for with either kernels or models that are truncated too low. Figure  shows the atmospheric temperature anomalies simulated in atmosphere-only simulations from CMIP6 models participating in RFMIP-ERF Tier 1 experiments for a 30-year time-slice simulation where CO2 concentrations are quadrupled relative to a pre-industrial control (piClim-4×CO2). Stratospheric cooling continues to increase above 10 hPa in all models where data are available. A similar pattern of stratospheric cooling occurs in the piClim-ghg experiment which evaluates forcing from present-day greenhouse gases (not shown). Standard CMIP6 diagnostics call for model output on 19 pressure levels: 1000, 925, 850, 700, 600, 500, 400, 300, 250, 200, 150, 100, 70, 50, 30, 20, 10, 5 and 1 hPa, whereas in CMIP5 the standard set of 17 pressure levels did not include 5 and 1 hPa. Therefore, CMIP5 models were missing important additional stratospheric cooling where kernels were used for adjustment calculations.

The truncation of stratospheric height in “low-top” radiative kernels (those with a top level lower in altitude than 1 hPa) has substantial consequences for adjustments to a greenhouse gas forcing. Figure  shows the stratospheric temperature adjustment for 4×CO2 in 13 models contributing to RFMIP. A simplified tropopause definition is used here, borrowed from , of a linear-in-latitude ramp from 100 hPa at the Equator to 300 hPa at the poles. There is a spread of around 1 W m-2 in calculated stratospheric temperature adjustment for each model using the full range of kernels, which is about 13 % of the effective radiative forcing (ERF) for a quadrupling of CO2 from these models . It can be seen in Fig.  that the kernel estimates for stratospheric adjustment to CO2 forcing are clustered into two groups and one high outlier for most models. The one model where kernel estimates are not clearly separated into high and low clusters is the GFDL-CM4 model, for which data are missing at 1 hPa. The low-top radiative kernels, with the exception of GFDL and ECHAM5, produce substantially lower estimates of the stratospheric temperature adjustment than the “high-top” kernels (HadGEM3-GA7.1, ECHAM6 and ECMWF-Oslo). The high outlier, ECMWF-RRTM, has a large flux change at the 1 hPa level. We show in Sect.  that adjustments calculated using the HadGEM3-GA7.1 kernel in the IPSL-CM6A-LR model for a quadrupled CO2 experiment provide small residuals (i.e. the adjustments are appropriately captured), suggesting that assuming there are no compensating errors, low-top kernels would underestimate the stratospheric temperature response and produce larger residuals.

To investigate the source of the differences between kernels in more detail, Fig.  shows the vertically integrated kernels from 1 hPa, 10 hPa, the full stratosphere (, definition) and the full atmosphere. As previously, kernels that do not include the uppermost layers have been extended based on the highest layer for which data are reported. Figure a gives the temperature adjustment from a uniform 1 K increase throughout the atmosphere. Here, there is little variation between kernels, and much of the total adjustment comes from the troposphere where the bulk of the atmosphere is present. Notably, the whole atmosphere adjustment to a 1 K temperature increase is of the order -3 W m-2, approximating (if slightly underestimating) the Planck feedback (-3.2 W m-2 K-1).

Figure b shows the layer contributions when each kernel is applied to the atmospheric profile from the IPSL-CM6A-LR model for piClim-4×CO2. In contrast to the isothermal case, the top 1 hPa, top 10 hPa and stratospheric temperature adjustments show substantial diversity between kernels. The large contributions from these layers follow from Fig. , where the 1 and 10 hPa layers are of the order 10 and 20 K cooler in the 4×CO2 run than in the control. The HadGEM3-GA7.1, GFDL, ECMWF-Oslo, ECMWF-RRTM and ECHAM6 kernels have a larger adjustment from the top 10 hPa than the other kernels. Despite being nominally a low-top kernel, the GFDL kernel has a similar magnitude and gradient of cooling between 10 and 5 hPa to the high-top kernels (Fig. ) and produces a similar result of total stratospheric adjustment to HadGEM3-GA7.1. The ECHAM5 kernel has more cooling around the 100 hPa level than any other kernel (Fig. ) and also has stratospheric adjustments that are similar to HadGEM3-GA7.1.

The differences between models for the top 10 hPa propagate through to the tropopause level and whole atmosphere. Differences in tropospheric adjustments between kernels (whole atmosphere minus stratosphere) are small, showing that choice of atmospheric base state and radiative transfer is not critical for tropospheric temperature adjustments. This analysis gives further confidence that the choice of radiative kernel is not that important in climate feedback studies (assuming state changes are sufficiently small to remain linear; ) as stratospheric temperature differences are small when differencing coupled atmosphere–ocean simulations and kernels show similar behaviour in the troposphere.

This section shows the linear behaviour of the HadGEM3-GA7.1 kernel used with IPSL-CM6A-LR climate model output. IPSL-CM6A-LR is chosen as all required diagnostics are available in this model to evaluate linearity, including estimates of the IRF from double calls, and ISCCP simulator diagnostics for clouds. This model is also representative of the RFMIP population, with 4×CO2 ERF and adjustments close to the multi-model average .

Linearity is evaluated by the size of the residual term, ϵ, which is any TOA flux changes not explained by instantaneous radiative forcing or kernel-calculated adjustments. A guideline of linearity for the kernel method is that the residual should be within 10 % of the ERF. We take two different approaches to calculate the residual. The first assumes that we have knowledge of the cloud adjustment term Ac (e.g. from the ISCCP simulator or NASA A-Train kernels convoluted with ISCCP simulated cloud output from climate models ), and knowledge of the IRF (Fi) from a double call in the online model. This is rare in practice, as double calls and ISCCP cloud diagnostics are not routinely archived on the Earth System Grid Federation (ESGF), the main distributed data source for CMIP6 output (although model participation is improving for ISCCP clouds). From this “perfect information” method we can calculate an all-sky residual ϵall as 6ϵall=F-Fi-ATa-ATs-Aq-Aα-Ac.

The second method does not require knowledge of the cloud adjustments but does require knowledge of the clear-sky IRF, and in this case the clear-sky residual term ϵclr can be calculated as 7ϵclr=Fclr-Ficlr-ATaclr-ATsclr-Aqclr-Aαclr. This clear-sky residual definition is more common in the literature , although often the clear-sky IRF is estimated rather than calculated directly. However, in some circumstances, clear-sky and all-sky IRF are known to be identically zero (e.g. in the LW spectrum to a change in the solar constant; ). In these cases, Eq. () can be used with Fi=Ficlr=0 to determine Ac, which is then plugged into Eq. () to calculate ϵall.

IPSL-CM6A-LR used two sets of double calls. In the RFMIP piClim-4×CO2 experiment (30-year time-slice atmosphere-only run with quadrupled CO2) the second radiation call saw a pre-industrial CO2 concentration. In the piClim-control experiment (pre-industrial atmosphere-only run) the second radiation call saw 4×CO2. The resulting IRF depends on the direction of the double call and is related to the underlying 4×CO2 or pre-industrial climatology. The 4×CO2 climatology that sees a pre-industrial second radiation call results in an IRF that is 1.26 W m-2 greater than the pre-industrial climatology that sees a 4×CO2 second radiation call (Table ). We take the mean of the two simulations to be the IRF.

Table  shows ERF, IRF, adjustments and residuals using the HadGEM3-GA7.1 radiative kernel with the IPSL-CM6A-LR model output and cloud adjustments from the ISCCP simulator kernel. We use ISCCP kernel values in the calculation of ϵall and also compare the ISCCP values to the cloud-masking estimate of cloud adjustment from Eq. (). For LW forcing the residuals are 0.28 W m-2 for ϵclr and 0.38 W m-2 for ϵall. Residuals are present possibly due to a slight breakdown in the linearity assumption for a forcing as large as 4×CO2 ; however, the residuals are comfortably within the 10 % linearity guideline. SW residuals are also within 10 % of the ERF, with ϵclr being particularly small. For the net fluxes, forcings add but residuals partly cancel, such that ϵclr and ϵall are 3.2 % and 1.9 % of the ERF respectively.

Our results in Table  can be compared with the results of for 11 CMIP5 models. The instantaneous forcing and tropospheric adjustments from IPSL-CM6A-LR with the HadGEM3-GA7.1 kernel are in line with the CMIP5 forcing and adjustments except for the stratospheric temperature adjustment, which is outside the 2σ range from CMIP5 models (Table ). As discussed in , the 4×CO2 ERF in available CMIP6 models (7.98 W m-2) is (non-significantly) greater than in corresponding CMIP5 sstClim4×CO2 experiments (7.53 W m-2), which is also the case for IPSL-CM6A-LR in our comparison to . IPSL-CM6A-LR is near the centre of the CMIP6 range for stratospheric temperature adjustment and so is a representative model of this ensemble. This could suggest stratospheric temperature adjustment increase as one driver of the increase in ERF between CMIP5 and CMIP6 models, although as most CMIP5 output is only on 17 model levels up to 10 hPa a formal comparison is difficult.

The stratospheric temperature adjustment is the only adjustment estimate that varies significantly between radiative kernels . If a kernel that did not resolve stratospheric temperature adjustment well was used instead of HadGEM3-GA7.1, this adjustment (ATa strat. in Table ) would be smaller, and the overall residuals for LW and net responses to 4×CO2 would be larger. From Fig.  it can be seen that some kernels produce a stratospheric temperature adjustment around 0.7 W m-2 lower than the HadGEM3-GA7.1 kernel, which would lead to residuals of the order 1 W m-2 using these kernels, or more than 10 % of the ERF.

The HadGEM3-GA7.1 radiative kernels are available at 10.5281/zenodo.3594673 .

This paper serves two purposes – it introduces the radiative kernel based on the high-top HadGEM3-GA7.1 general circulation model, and it compares estimates of the stratospheric temperature adjustment obtained with a variety of different radiative kernels for quadrupled CO2 experiments. The HadGEM3-GA7.1 kernel is the first to our knowledge that has been produced using a CMIP6-era model, with a focus on the 19 pressure level diagnostics available in CMIP6 output, although other kernels in the literature have used high-top atmospheric profiles . Radiative kernels are produced for both top-of-atmosphere and surface fluxes and are available on the native 85-level hybrid height grid in addition to the 19 CMIP6 pressure levels.

We show that there is a significant diversity, of about 1 W m-2 or 13 % of the ERF for a quadrupling of CO2, for estimates of stratospheric temperature adjustments to CO2 depending on the radiative kernel used to derive the estimate. As tropospheric and land surface adjustments vary little between kernels to a variety of different forcing agents , these differences in stratospheric temperature adjustments lead to differing estimates of the total adjustment, and also of the IRF if it is calculated as a residual . Climate feedbacks are little affected by the choice of kernel, due to the fact that stratospheric temperatures readjust quickly to an imposed forcing in coupled model simulations .

While only one model (IPSL-CM6A-LR) archived IRF from a double call and a rigorous multi-model test is not possible, we show that the HadGEM3-GA7.1 kernel diagnoses IRF and adjustments with a small residual owing to the increased stratospheric resolution available compared to many CMIP3- and CMIP5-era kernels. We suggest that radiative kernels with a higher stratospheric resolution and model top are better able to fully capture stratospheric adjustments to CO2 forcing in general and generate smaller residuals. This effect has become more prominent with the additional 5 and 1 hPa model levels archived as standard in processed CMIP6 model output compared to CMIP5. Archiving instantaneous radiative forcing from more models would be beneficial to further test the linearity assumption of the radiative kernel method.