Abstract

A vertically nonuniform warming of the troposphere yields a lapse rate feedback by altering the infrared irradiance to space relative to that of a vertically uniform tropospheric warming. The lapse rate feedback is negative at low latitudes, as a result of moist convective processes, and positive at high latitudes, due to stable stratification conditions that effectively trap warming near the surface. It is shown that this feedback pattern leads to polar amplification of the temperature response induced by a radiative forcing. The results are obtained by suppressing the lapse rate feedback in the Community Climate System Model, version 4 (CCSM4). The lapse rate feedback accounts for 15% of the Arctic amplification and 20% of the amplification in the Antarctic region. The fraction of the amplification that can be attributed to the surface albedo feedback, associated with melting of snow and ice, is 40% in the Arctic and 65% in Antarctica. It is further found that the surface albedo and lapse rate feedbacks interact considerably at high latitudes to the extent that they cannot be considered independent feedback mechanisms at the global scale.

1. Introduction

A forcing of the climate system due to a radiative imbalance at the top of the atmosphere (TOA) results in a change of Earth’s surface temperatures. This temperature response may activate feedback processes within the climate system that either further enhance or dampen the TOA imbalance, and hereby further increase or decrease the temperature response.

The temperature response will not be constant over Earth’s surface but tends to be larger at higher than at lower latitudes, which is referred to as polar temperature amplification (Manabe and Wetherald 1975; Hansen et al. 2005; Holland and Bitz 2003). Observations reveal that the ongoing global warming is amplified in the Arctic (Serreze and Francis 2006; Graversen et al. 2008; Serreze and Barry 2011). The amplification may be due to the forcing and feedback processes being stronger at high latitudes. But it may also be a consequence of other processes that redistribute energy within the climate system, and that are modified due to the forcing. For example, changes of the ocean and atmospheric energy transport in response to a forcing induce warming at some latitudes and cooling at others (Graversen 2006). Although such changes can have large local effects, they constitute only weak global radiative feedbacks as they contribute little to the global-mean TOA imbalance. Many studies find that the meridional atmospheric energy transport does not contribute to polar amplification but rather damps it (Hwang et al. 2011; Koenigk et al. 2013; Skific and Francis 2013), whereas other studies point in the opposite direction (Alexeev et al. 2005; Langen and Alexeev 2007; Graversen et al. 2008). In fact, the sign of the energy transport is model dependent; models with weak Arctic warming tend to exhibit an increased transport, and vice versa (Pithan and Mauritsen 2014).

Forcings associated with a change of the atmospheric content of CO2 or a change of the solar constant do not directly contribute to polar temperature amplification, since the induced radiative imbalance at TOA is larger at lower than at high latitudes (Hansen et al. 2005). In contrast, some feedback processes are believed to contribute to the amplification—for instance, the surface albedo feedback (Arrhenius 1896) and the lapse rate feedback (Manabe and Wetherald 1975), both of which are examined here. The water vapor feedback, which strongly enhances the global temperature response (Arrhenius 1896; Held and Soden 2000), seems most active at low latitudes (Langen et al. 2012), whereas the feedback associated with cloud changes is more uncertain.

Although the tropospheric temperature will change in response to a forcing, the change may not be constant with height. The difference in temperature change with height gives rise to the lapse rate feedback. Because of the Clausius–Clapeyron relationship between temperature and water vapor saturation pressure, the saturated mixing ratio of water vapor increases more at lower than at upper levels in the troposphere when Earth is warming. In regions where strong convection is present, such as at tropical latitudes, this leads to an increase of latent heat release and warming of the upper troposphere (Hansen et al. 1984), which results in enhanced radiation back to space, and in a more efficient cooling of Earth. This contributes to a negative lapse rate feedback. At the high latitudes, stable stratification conditions in the lower troposphere result in a larger warming of the near-surface air than of the upper troposphere (Manabe and Wetherald 1975), which contributes to a regionally positive lapse rate feedback. Hence the lapse rate feedback is believed to be negative at low and positive at high latitudes, which leads to Arctic amplification (Pithan and Mauritsen 2014).

In the present study the lapse rate and the surface albedo feedback are suppressed by locking the lapse rate and the surface albedo one by one and in combination in a state-of-the-art climate model. Hereby the full effect on the climate system due to each of the feedbacks is investigated. The contribution from these feedbacks to the polar amplification and the interactions between the feedbacks are studied. Feedback parameters that are assumed to be unique for each feedback (Hansen et al. 1984) are estimated directly, as an alternative approach to the indirect offline methods, such as the more commonly used partial radiative perturbation (PRP) method (Wetherald and Manabe 1988) and the radiative kernel method (Soden et al. 2008).

In the PRP approach, the radiation code is run in a standalone mode with fields associated with a given feedback changed to a perturbation level and all other fields kept to a control level. For instance, the albedo feedback can be estimated by running the radiation code with surface albedo from a 2×CO2 climate and all other fields from a 1×CO2 climate. The kernel method is similar to the PRP approach, but here a unit perturbation of the fields associated with a given feedback is considered in order to obtain what is referred to as radiative kernels. These kernels can be multiplied by the actual perturbation of the fields in question, normalized by the temperature change at the surface, in order to obtain the feedback parameter. An advantage of the kernel method is that the radiative kernels are to some extent independent of models (Soden et al.2008); when the kernels are already determined, it is straightforward and computationally inexpensive to calculate the feedback parameters.

The online method wherein feedbacks are suppressed in climate models has been applied earlier in order to study the surface albedo feedback (Hall 2004; Graversen and Wang 2009; Mauritsen et al. 2013), the water vapor feedback (Schneider et al. 1999; Hall and Manabe 1999; Langen et al. 2012; Mauritsen et al. 2013), and the cloud feedback (Vavrus 2004; Langen et al. 2012; Mauritsen et al. 2013). In addition to studying the effect of the feedbacks, Langen et al. (2012) and Mauritsen et al. (2013) also tested the notion that the total temperature change can be split into parts that can be attributed to each of the feedbacks. Both studies concluded that this was indeed the case for the feedbacks considered.

In the present study we lock the lapse rate and the surface albedo feedback. As far as the authors know, it is the first time the lapse rate feedback has been suppressed in a climate model.

2. Model description

The Community Climate System Model, version 4 (CCSM4; Gent et al. 2011), from the National Center for Atmospheric Research (NCAR) is used. The model system includes submodels for the atmosphere, land surface processes, sea ice, and ocean. The atmosphere model has a finite-volume dynamical core with 26 vertical layers and ~2° horizontal resolution. It is run both in a slab-ocean mode (SOM) and a data-ocean mode (DOM), where in the latter the sea surface temperatures and the sea ice are fixed to a climatology. In the SOM version, the ocean part includes an isothermal mixed layer only, which has a fixed horizontal transport of energy. The horizontal ocean energy transports in the slab-ocean model, often referred to as the q fluxes, are determined from the climatology of an equilibrium run including a full dynamical ocean model (Bitz et al. 2012). The q fluxes are given as the climatological mean of the imbalance between the ocean mixed-layer energy change, and the energy flux into the ocean across the atmosphere–ocean and ice–ocean interface and from river discharge. The q fluxes as well as the depth of the mixed layer were based on the climatology from the last 45 years of a 560-yr control run. For the DOM version, the sea surface temperatures (SSTs) and sea ice are taken from the same 45-yr coupled-model climatology. Annual means of the globally averaged surface air temperature (SAT) and sea ice extent from the coupled model and the DOM control runs are shown in Fig. 1 along with the results from all SOM experiments.

Fig. 1.

Globally averaged, annual-mean (a) SAT and (b) sea ice extent as a function of model year for all slab-ocean experiments, the ocean–atmosphere coupled control run, and the data ocean control run with fixed sea surface temperatures and sea ice (SST/SI). The slab-ocean experiments are shown both for a 1×CO2 and 2×CO2 climate for the free experiment, the SAT-based locked LR, the TrMT-based locked LR, the locked SA, and the experiment with both SA and SAT-based LR locked.

Fig. 1.

Globally averaged, annual-mean (a) SAT and (b) sea ice extent as a function of model year for all slab-ocean experiments, the ocean–atmosphere coupled control run, and the data ocean control run with fixed sea surface temperatures and sea ice (SST/SI). The slab-ocean experiments are shown both for a 1×CO2 and 2×CO2 climate for the free experiment, the SAT-based locked LR, the TrMT-based locked LR, the locked SA, and the experiment with both SA and SAT-based LR locked.

The model is run in a 1×CO2 and a 2×CO2 configuration with the atmospheric CO2 level set to 284.7 and 569.4 ppm, respectively, where the former represents the preindustrial conditions. In these two configurations, experiments are undertaken with locked lapse rate (LR), with locked surface albedo (SA), with both locked, or with both free.

3. Experiments

The surface albedo and the lapse rate feedbacks are investigated by designing model versions where a given feedback is suppressed. For each version the climate response to a CO2 doubling is estimated. Subsequently the response is compared between versions with a feedback suppressed and versions where it is included, in order to investigate the feedback in question.

In Fig. 1 time series of the global-mean and annual-mean SAT and sea ice extent are shown for all SOM experiments as well as for the coupled ocean model and the DOM control runs. The 1×CO2 experiments with locked lapse rate feedback are initiated at year 20 from corresponding experiments with the feedback active. In the 2×CO2 experiments, the CO2 is doubled instantaneously at the beginning of the year 30. The 1×CO2 locked lapse rate experiments drift up to 1 K relative to the control run, whereas the locked surface albedo experiment shows negligible drift. Note that the climate response in a world with the lapse rate feedback suppressed is estimated by comparing the 2×CO2 and 1×CO2 experiments that both have the lapse rate locked. The drift of the suppressed lapse rate experiments relative to the free experiments has no impact on the results as long as the drift is the same in both the 1×CO2 and the 2×CO2 experiments, which is assumed in the present study. The experiments are investigated based on 80-yr climatologies over the years 81–160.

All experiments are repeated in a DOM configuration. These experiments are run for 70 years and 50-yr climatologies are taken over the years 21–70. The 1×CO2 DOM control experiment is shown Fig. 1. The nomenclature of the experiments indicated in the legend of the figure is used throughout this paper.

a. Locking of the lapse rate

The lapse rate feedback is suppressed by locking the tropospheric lapse rate in the longwave radiation code. In the present study, two ways of regarding the lapse rate feedback are investigated, the tropospheric mean temperature (TrMT)-based feedback and the SAT-based feedback.

Variations in the vertical distribution of the tropospheric warming impact the atmospheric radiation to space and to the surface. Hereby the tropospheric warming structure influences the atmospheric cooling ability and the warming of the surface. In the polar regions stable stratification conditions often prevail. This hampers vertical mixing of the troposphere and implies that a forcing-induced energy input at the surface leads to an amplified warming of the near-surface troposphere relative to the upper tropospheric levels. In comparison to a vertically homogeneous warming, such an inhomogeneous warming structure causes a different cooling ability of the atmosphere by modifying the atmospheric radiation to the surface and to space. In addition, in the polar regions the inhomogeneous warming structure contributes to surface warming, since this structure compared to a vertically homogeneous warming induces more radiation toward the surface. Here we explore the feedback associated with the tropospheric warming being distributed inhomogeneously with height. We explore the feedback response in terms of atmospheric radiative cooling and surface warming. This TrMT-based lapse rate feedback is suppressed online in the model by substituting the longwave radiation to space and at the surface with estimates based on temperature profiles, where the temperature changes at all levels in the troposphere are equal to the tropospheric mean change.

The SAT-based feedback is the traditional way of regarding the feedback. Here we explore the feedback associated with tropospheric temperature changes being different from the temperature change at the surface. The feedback is considered solely as a radiative feedback. Radiative feedbacks cause an alteration of the energy balance at TOA. Therefore, when suppressing the SAT-based lapse rate feedback, only the longwave radiation to space and not that to the surface is substituted.

These are two ways of regarding the lapse rate feedback; many others may be defined. The definition of the SAT-based feedback applied here is consistent with other studies and the results can therefore be compared. Note that a fundamental difference between the approaches is that in the SAT-based suppression of the feedback, the tropospheric warming is locked to the warming at the surface, whereas in the TrMT approach the troposphere may warm or cool freely relative to the surface, but in a vertically homogeneous manner. In the TrMT approach, the longwave flux to the surface is estimated from the locked lapse rate profiles, whereas in the SAT approach it is not. This is because the TrMT approach is designed to suppress the feedback between the surface and the atmosphere, while the SAT-based lapse rate feedback is regarded in a radiative feedback context; for that reason only the TOA fluxes are relevant.

1) The SAT-based feedback

For the purpose of suppressing the lapse rate feedback using the SAT-based method, longwave radiation to space is calculated from locked lapse rate temperature profiles using the longwave radiation model code. In these profiles, the temperature changes at the tropospheric levels are set equal to the SAT changes, whereby the lapse rate in the troposphere is held fixed. The locked lapse rate is obtained by adding the SAT change to the climatological temperature profile:

 
formula

where Tlock(z) is the temperature profile with a locked lapse rate, Tclim(z) and TSclim are 1×CO2 climatologies of the temperature profile and SAT, respectively, and TS is the SAT simulated online by the model. All fields are functions of horizontal grid points and time, and z indicates height. The climatological fields are from a 70-yr climatology of a slab-ocean control run with all feedbacks active and a 1×CO2 climate. These have an hourly resolution that corresponds to the frequency of the call to the radiation code. Hence Tclim(z) and TSclim include one year of data and are functions of the day of the year, and the hour of the day.

In practice the longwave radiation code is run twice per radiation call; first in its original form using the free temperature profile simulated by the model T(z), and second with T(z) substituted by Tlock(z). From the two calls, the TOA longwave flux difference is determined. This energy difference constitutes the effect on TOA radiation associated with the locked lapse rate. The longwave flux to space is estimated by the model for diagnostic purposes only. To take the energy difference into account, and hereby suppressing the lapse rate feedback, an atmospheric warming rate, Q(z), is implemented between ps and ptr, the pressure at the surface and at the tropopause, respectively. The warming rate, Q(z), is chosen to be constant over the tropospheric levels, whereby, the dry-static stability of the troposphere is unaffected by Q(z). The vertical integral of equals the TOA longwave radiation difference:

 
formula

Here Lt and is longwave radiation up at TOA based on the free and locked temperature profiles, respectively, and the brackets 〈〉 indicate the mass-weighted vertical average over the troposphere:

 
formula

Again all fields are functions of horizontal grid points and time.

For the tropopause height, a definition from the World Meteorological Organization (WMO) is applied (Reichler et al. 2003): The tropopause is encountered at the lowest level where −dT/dz < 2 K km−1, and where the average lapse rate over the following 2 km does not exceed 2 K km−1. In a few cases, especially over Antarctica, the tropopause cannot be determined using this definition. The tropopause height is then taken from the 70-yr climatology from the slab-ocean control experiment mentioned above. This backup tropopause climatology has a monthly resolution. By implementing the tropopause height this way it can vary in time and respond to an atmospheric CO2 doubling. Zonal averages of the tropopause height from the control experiment where all feedbacks are active are shown in Fig. 2 for both the 1×CO2 and 2×CO2 case along with the temperature field from the 1×CO2 experiment.

Fig. 2.

Zonal-mean and annual-mean climatologies of the tropopause height in the 1×CO2 (red line) and the 2×CO2 (white line) free experiment as a function of latitude. Also displayed is the climatology of the atmospheric temperatures from the 1×CO2 climate (shading) as a function of latitude and pressure.

Fig. 2.

Zonal-mean and annual-mean climatologies of the tropopause height in the 1×CO2 (red line) and the 2×CO2 (white line) free experiment as a function of latitude. Also displayed is the climatology of the atmospheric temperatures from the 1×CO2 climate (shading) as a function of latitude and pressure.

In summary, in this design of suppressing the lapse rate feedback, the only online change to the model is a shift of the energy with a constant amount over the tropospheric column. The point is that this energy shift offsets the TOA energy flux associated with the lapse rate feedback. Note that the radiative warming/cooling rates at each atmospheric level associated with the model temperature profiles are still used online in the model. In principle the lapse rate feedback may be suppressed by taking these warming/cooling rates from the locked lapse rate profiles instead of implementing Q(z). However, this approach would likely have a large undesired effects on other variables important for the climate such as water vapor and clouds, which is not the intention.

2) The TrMT-based feedback

The TrMT-based method is similar to that imposed for the SAT-based feedback with three exceptions. First, the locked lapse rate is obtained by adding the mass-weighted mean change of the tropospheric temperatures to the climatological temperature profile:

 
formula

Second, both the TOA upward and the surface downward longwave radiation from the atmosphere with the locked temperature profile are taken into account. Hence the downward radiation is taken from the radiation call with the locked temperature profile and used online in the model. Third, the difference in surface longwave radiation between the two radiation calculations is now also taken into account when estimating the atmospheric warming rates, Q(z):

 
formula

where Ls is longwave radiation downward at the surface. Again Q(z) is chosen to be constant with height.

b. Locking of the surface albedo

The surface albedo feedback is suppressed by locking the surface albedo to the 70-yr climatology from a preindustrial control experiment. This climatology was also used for locking of the lapse rate feedback. An hourly climatology of the surface albedo is used, which corresponds to the frequency of the call to the shortwave radiation code. Locking of the surface albedo has been done previously (Hall 2004; Bitz 2008; Graversen and Wang 2009; Mauritsen et al. 2013).

The basic idea is that the albedo is kept fixed to a climatology even though the surface properties change in a way that would alter the albedo. For instance, if sea ice appears in a certain grid point and time of the year, but the climatology indicates no sea ice here, the locked surface albedo used in the model’s radiation calculations will attain that of the ocean.

The albedo in CCSM4 distinguishes between shortwave radiation with wavelength larger and smaller than 0.7 μm, and between direct and diffuse radiation, which results in four albedo fields. In the atmospheric model component, the surface albedo is locked by substituting in the shortwave radiation code the albedo fields from the model with those from the climatology. This code estimates the shortwave heating rates as well as the shortwave fluxes at the top and bottom of the atmosphere. However, the actual shortwave absorption by the surface in the model is not taken from the atmosphere radiation code. Instead, this is estimated separately in each of the surface model components for land, sea ice, and ocean based on the downward shortwave fluxes at the surface and based on the surface properties.

In the sea ice model, the grid cells are separated into five ice categories. Each ice category is further divided into bare ice, snow-covered ice, and ice with ponds. Hence the ice-covered part of the grid cell is constituted by 15 minor parts in which shortwave absorption is estimated separately. The absorption is estimated based on the delta-Eddington method (Briegleb and Light 2007). Each of the 15 minor parts includes a number of vertical snow and ice layers in which shortwave absorption is calculated. Also a fraction of the incoming solar radiation passes through the snow and ice and is absorbed by the ocean beneath.

The locking of the surface albedo is implemented by scaling the absorption so that the albedo equals the climatological albedo. A scaling coefficient c is obtained by dividing S = l + Al with S = f + Af:

 
formula

Here S is downward shortwave radiation at the surface, and Af and Al are the total surface shortwave absorption associated with the modeled and locked surface albedo, αf and αl, respectively. The scaling coefficient can now be applied to the absorption in each of the layers j, yielding , so that

 
formula

where N includes all snow and ice layers as well as the radiation fraction that is passed to the ocean beneath. Note that the scaling coefficient is estimated separately for each of the 15 ice parts, and for each of the four albedo types.

In the land model, absorption is taken into account both for the canopy and for the ground. The model includes urban areas where absorption is encountered separately (e.g., roofs, roads, and house walls). A method analog to that from the ice model, Eq. (3), is used to scale the absorption in order to keep the albedo locked to the climatology. For the ocean the modeled albedo is simply substituted with the climatological albedo.

The effect on shortwave radiation due to locking of the surface albedo is shown in Fig. 3. Here differences between the 2×CO2 and the 1×CO2 climatology are shown. In the free experiment, including the surface albedo feedback, the net shortwave radiation at TOA increases considerably at the high latitudes in response to the albedo change associated with the retreat of snow and sea ice. The clear-sky fluxes indicate that this increase at high latitudes would have been much stronger, had it not been for the masking effect of clouds.

Fig. 3.

Doubling-of-CO2 change of net shortwave radiation for all sky at TOA (solid line), for clear sky at TOA (thick dotted line), and for clear sky at surface (thin dotted line), for (a) the free experiment and (b) the locked SA experiment where the surface albedo feedback is suppressed. All fluxes are positive downward and are estimated as the difference between the 2×CO2 and 1×CO2 climatologies.

Fig. 3.

Doubling-of-CO2 change of net shortwave radiation for all sky at TOA (solid line), for clear sky at TOA (thick dotted line), and for clear sky at surface (thin dotted line), for (a) the free experiment and (b) the locked SA experiment where the surface albedo feedback is suppressed. All fluxes are positive downward and are estimated as the difference between the 2×CO2 and 1×CO2 climatologies.

In the locked surface albedo experiment there is a small increase of net TOA radiation under clear-sky conditions, especially at high latitudes. This increase is due to the enhanced shortwave absorption by water vapor in the clear-sky atmosphere. The difference between the net clear-sky TOA radiation and the net clear-sky surface radiation reflects the increase in absorption by atmospheric gases due to the CO2 doubling. The increase is most pronounced at lower latitudes consistent with the increase of water vapor being largest here. The enhanced shortwave absorption by water vapor also leads to a decrease of shortwave radiation reaching the surface in the clear-sky atmosphere, which explains the surface clear-sky net shortwave radiation change being negative. Under clear-sky conditions the absorption of reflected solar radiation plays a larger role at the high latitudes than at the low latitudes. Although the effect is small, this is the reason for the increase in net clear-sky shortwave radiation at TOA being larger at the polar latitudes than farther equatorward in the experiment with the surface albedo feedback suppressed.

4. Lapse rate feedback

Profiles of atmospheric temperature changes due to a CO2 doubling are shown in Fig. 4 for both types of locking of the lapse rate. Profiles are shown for the polar and equatorial regions separately. The changes of the locked lapse rate profiles, based on Eqs. (1) and (2), are indicated by dotted lines, while the changes of the free lapse rate profiles are shown by solid lines.

Fig. 4.

Area-mean and annual-mean profiles of atmospheric temperature changes due to a CO2 doubling. Shown are changes of the free temperature profiles in the free experiment with all feedbacks active (thin solid lines) and in the experiment with the lapse rate feedback suppressed (thick lines). Also shown are the changes of the locked temperature profiles used in the longwave radiation code in the experiment with the lapse rate feedback suppressed (dotted lines): (a)–(c) SAT-based locking of the temperature profile, and (d)–(f) TrMT-based locking. Hence (a)–(c) show results from the locked LR (SAT) and (d)–(f) from the locked LR (TrMT) experiment, while results from the free experiment are shown in all panels. Profiles are shown for (a),(d) the Arctic (north of 60°N), (b),(e) the equatorial region (5°S–5°N), and (c),(f) the Antarctic region (south of 60°S). In some panels, the thin solid line underlies the thick line.

Fig. 4.

Area-mean and annual-mean profiles of atmospheric temperature changes due to a CO2 doubling. Shown are changes of the free temperature profiles in the free experiment with all feedbacks active (thin solid lines) and in the experiment with the lapse rate feedback suppressed (thick lines). Also shown are the changes of the locked temperature profiles used in the longwave radiation code in the experiment with the lapse rate feedback suppressed (dotted lines): (a)–(c) SAT-based locking of the temperature profile, and (d)–(f) TrMT-based locking. Hence (a)–(c) show results from the locked LR (SAT) and (d)–(f) from the locked LR (TrMT) experiment, while results from the free experiment are shown in all panels. Profiles are shown for (a),(d) the Arctic (north of 60°N), (b),(e) the equatorial region (5°S–5°N), and (c),(f) the Antarctic region (south of 60°S). In some panels, the thin solid line underlies the thick line.

For the SAT-based locking at the equatorial latitudes, the changes of the locked tropospheric temperatures are smaller than the changes of the free temperatures, while the opposite situation prevails at the polar latitudes. This is evident from a comparison of the thick solid and the dotted line in Figs. 4a–c. Since radiation to space is based on the locked profiles, the radiative cooling at TOA is reduced at low latitudes and increased at high. This is evident from Fig. 5a, where the difference in outgoing longwave radiation between the locked and the free temperature profiles are indicated by solid lines. The corresponding differences of longwave radiation at the surface are indicated by dashed lines. The differences are shown as a function of latitude. The thick solid line in Fig. 5a reveals that the lapse rate feedback induces an increase in radiation to space at low latitudes and a decrease at high latitudes. This is consistent with the temperature changes at upper tropospheric levels at the equatorial latitudes being smaller for the locked than for the free profiles, and vice versa at the polar latitudes (Fig. 4).

Fig. 5.

The lapse rate response in longwave radiation upward at the TOA (solid lines) and downward at the surface (dotted lines), for the experiment where the lapse rate feedback is included (thin lines) and where it is suppressed (thick lines). These responses are estimated as the difference between the free lapse rate and locked lapse rate doubling-of-CO2 longwave radiation change upward at TOA and downward at the surface, respectively. The quantities are shown both for the (a) SAT-based and (b) TrMT-based lapse rate feedback. Hence (a) shows results from the locked LR (SAT) and (b) for the locked LR (TrMT) experiment, while results from the free experiment are shown in all panels. In (b) the thin solid line is underlying the thick solid line.

Fig. 5.

The lapse rate response in longwave radiation upward at the TOA (solid lines) and downward at the surface (dotted lines), for the experiment where the lapse rate feedback is included (thin lines) and where it is suppressed (thick lines). These responses are estimated as the difference between the free lapse rate and locked lapse rate doubling-of-CO2 longwave radiation change upward at TOA and downward at the surface, respectively. The quantities are shown both for the (a) SAT-based and (b) TrMT-based lapse rate feedback. Hence (a) shows results from the locked LR (SAT) and (b) for the locked LR (TrMT) experiment, while results from the free experiment are shown in all panels. In (b) the thin solid line is underlying the thick solid line.

In the upper troposphere, the TrMT-based locking of the lapse rate shows the same pattern of differences as that of the SAT-based locking, although the differences are smaller for the TrMT-based locking (Fig. 4). However, in the lower troposphere the pattern is the opposite between the two types of locking: At the equatorial latitudes the changes of the locked profiles are larger than those of the free profiles for the TrMT-based feedback and vice versa for the SAT-based feedback, whereas the opposite situation prevails in the polar areas. These differences between the two types of locking are reflected in the impact on the radiation: A comparison of the solid line in Figs. 5a and 5b indicates that the effect on radiation to space by the lapse rate feedback is much smaller for the TrMT-based than for the SAT-based feedback. A comparison of the dashed lines shows that the effect on surface radiation is reversed between the two types. Hence the two types of locking the feedback are rather different when it comes to the impact on radiation. However, as will be shown later, both feedbacks cause polar temperature amplification by about the same magnitude. For the SAT-based feedback, it is the increase in radiation to space at low latitudes, and the decrease at high latitudes that eventually lead to polar amplification. For the TrMT-based feedback, polar amplification is mostly a result of the enhanced radiation to the surface at high latitudes and the reduction at low latitudes.

The thin and the thick solid lines in Fig. 4 show the change of tropospheric temperatures for experiments with the feedback active and suppressed, respectively. Likewise thin and thick lines in Fig. 5 show radiation responses due to the feedback for the active and suppressed feedback experiments. Note that for the free experiments, the locked profiles are not used online in the model but are estimated for diagnostic purposes only.

The SAT-based feedback leads to a reduction in its effect on TOA radiation as it becomes active (Fig. 5a). This is due to the cooling effect of the feedback at low latitudes, which is most efficient in the upper troposphere (Fig. 4b). At high latitudes, in terms of TOA radiation this feedback strengthens as it becomes active. This is a result of the warming associated with the feedback being larger at the surface than in the upper troposphere, which enhances the difference between the free and the locked temperatures in the upper troposphere.

The TrMT-based feedback tends to amplify itself in terms of radiation to the surface in the polar areas, since the radiation to the surface associated with this feedback is larger when the feedback is active than when it is suppressed (Fig. 5b). This is because the radiation to the surface associated with this feedback causes polar surface warming, which enhances the difference between the free and the locked temperature changes close to the surface.

5. Polar amplification

Both the lapse rate and the surface albedo feedbacks induce polar amplification associated with a CO2 forcing, which is evident in Fig. 6a. This figure shows annual-mean SAT change due to a doubling of CO2 for different experiments as a function of latitude. For all experiments polar amplification is evident in all seasons except in the Arctic during boreal summer and around Antarctica during austral summer, when melting of sea ice hampers the surface warming (not shown).

Fig. 6.

Annual-mean and zonal-mean doubling-of-CO2 SAT response. (a) The response is shown for the free experiment including all feedbacks (red line), for the locked SAT-based lapse rate (dark green line), for the locked TrMT-based lapse rate (light green line), and for the locked surface albedo experiment (blue line). (b) The difference in responses between the free experiment and each of the locked experiments (free minus locked) is shown. Lines at the bottom at (b) indicate latitudes where the difference between the free and the locked experiments are significant on the 95% level. See  appendix B concerning the method used to estimate the significances.

Fig. 6.

Annual-mean and zonal-mean doubling-of-CO2 SAT response. (a) The response is shown for the free experiment including all feedbacks (red line), for the locked SAT-based lapse rate (dark green line), for the locked TrMT-based lapse rate (light green line), and for the locked surface albedo experiment (blue line). (b) The difference in responses between the free experiment and each of the locked experiments (free minus locked) is shown. Lines at the bottom at (b) indicate latitudes where the difference between the free and the locked experiments are significant on the 95% level. See  appendix B concerning the method used to estimate the significances.

The differences between the free feedback experiment and each of the other experiments, where a feedback mechanism has been locked, are shown in Fig. 6b along with an estimate of the statistical significance of the differences (see  appendix B for details concerning the estimation of the statistical significance). In general, the SAT-based lapse rate feedback (the dark green line) induces reduced warming at low but increased warming at high latitudes, hereby enhancing the polar amplification. This is consistent with the lapse rate feedback increasing the longwave radiation to space at low latitudes and decreasing it at high latitudes (Fig. 5).

The global-mean surface air temperature change for the different experiments is given by Fig. 7a along with the Arctic and Antarctic changes. The SAT-based lapse rate feedback shows only negligible impact on the global-mean change. However, since the high-latitude warming is larger when the lapse rate feedback is included, part of the polar amplification (PA) can be attributed to this feedback, around 15% for the Arctic and around 30% for the Antarctic (Fig. 7b). This relative contribution of a given feedback is obtained by

 
formula

where PAfree and PAlocked are the polar amplification in the experiment with active and suppressed feedback, respectively.

Fig. 7.

Polar amplification of SAT associated with a CO2 doubling: (a) the global, Arctic (north of 60°N), and Antarctic (south of 60°S) temperature change, and (b) the Arctic and Antarctic amplification defined as the polar divided by the global change. Shown are the free experiment including all feedbacks (red), the locked SAT-based lapse rate (dark green), the TrMT-based locked lapse rate (light green), the locked surface albedo (dark blue), and the experiment with both surface albedo and SAT-based lapse rate locked (light blue).

Fig. 7.

Polar amplification of SAT associated with a CO2 doubling: (a) the global, Arctic (north of 60°N), and Antarctic (south of 60°S) temperature change, and (b) the Arctic and Antarctic amplification defined as the polar divided by the global change. Shown are the free experiment including all feedbacks (red), the locked SAT-based lapse rate (dark green), the TrMT-based locked lapse rate (light green), the locked surface albedo (dark blue), and the experiment with both surface albedo and SAT-based lapse rate locked (light blue).

The TrMT-based lapse rate feedback (light green line in Fig. 6) shows roughly the same amplification pattern as that based on SAT, although the cooling is smaller at low latitudes, and the warming larger at high latitudes. This pattern emerges despite the fact that the two ways of regarding the feedback lead to rather different radiation patterns, both at the top and bottom of the atmosphere (Fig. 5). The TrMT-based feedback explains ~15% of the Arctic amplification and 20% of the amplification in the Antarctic region (Fig. 7b).

The surface albedo feedback enhances the warming at all latitudes although the high latitudes are the most affected. The temperature response to the albedo feedback is clearly larger than that resulting from the lapse rate feedback. As the albedo feedback becomes active, the global-mean temperature response due to a CO2 doubling increases by ~1 K, while the response in the Arctic increases by ~3 K and in the Antarctic region by 3.5 K. Hence in this CCSM4 slab-ocean model version, the albedo feedback explains a large part of the polar amplification, around 40% for the Arctic and 65% for the Antarctic. However, although the albedo feedback appears important, approximately half of the polar amplification is still unexplained by this feedback as given by the model employed here.

Even though the SAT-based lapse rate feedback contributes with little change of the global-mean temperature in the free-feedback experiment, it induces a cooling of around 0.5 K when the surface albedo feedback is not active. As will be discussed further in the next section, in the polar areas the lapse rate feedback becomes weaker when the surface albedo feedback is suppressed. This is due to the surface warming at high latitudes being smaller when the surface albedo feedback is inactive. Hence globally, the lapse rate feedback becomes stronger (more negative) as it becomes weaker at the high latitudes, where it is positive.

6. Feedback parameters

Here the lapse rate and surface albedo feedbacks are examined in the light of what is known as the feedback parameters, which are assumed to be unique numbers associated with each of the feedback processes. For a small forcing of the climate system F constituted by a radiative perturbation at the climate system boundaries, it may be assumed that the equilibrium surface air temperature response ΔTeq is proportional to the forcing:

 
formula

where f is a climate sensitivity factor. A basic reasoning behind this assumption is that although Earth’s radiation to space is dependent on temperature to the power of 4, small radiation perturbations of a few Watts per square meter to a good approximation can be assumed to depend linearly on temperature. Also for small perturbations, strong nonlinear feedback processes are less likely to be invoked. The size of f depends on feedback processes in the climate system that are activated by the climate change induced by the forcing. The major feedback process is that associated with water vapor changes, which is believed to roughly double the global-mean temperature response (e.g., Held and Soden 2000). Other important feedbacks are those of clouds, surface albedo, and lapse rate, where the latter two are examined here.

It may be further assumed that the total temperature response (ΔTeq) can be divided into constant fractions that can be attributed to each of the feedbacks, i:

 
formula

Then f is given as f = −1/λ (see  appendix A), where the feedback parameter λ consists of parts that are unique for each feedback:

 
formula

whereby Eq. (4) can be expressed:

 
formula

This convention implies that a positive (negative) feedback, enhancing (reducing) the temperature response, is associated with a positive (negative) λi. Because of the assumption given by Eq. (5), the feedback parameters are independent of each other (otherwise a fraction of the temperature change is attributed to more than one feedback). A split of ΔTeq into parts was studied for the water vapor and cloud feedbacks by Langen et al. (2012) and for the water vapor, cloud, and surface albedo feedback by Mauritsen et al. (2013), and a good agreement with Eq. (5) was found.

It may be shown that the assumption Eq. (5) is equivalent to the constraint that feedbacks (i ≥ 1) induce a radiation imbalance at the top of the atmosphere that is linearly dependent on the total temperature change, ΔTeq:

 
formula

where ΔRi is the feedback-induced TOA imbalance. Hence positive feedbacks increase the radiative imbalance at TOA induced by the forcing, whereas negative feedbacks reduce it.

Note that the feedbacks are invoked by the total temperature change and that the magnitude of the temperature response associated with a given feedback depends on the total temperature change (ΔTeq). Hence the magnitude of the temperature response that a feedback induces depends also on the contribution to ΔTeq from the other feedbacks. In the experiments examined in this study, where feedbacks are suppressed, the associated temperature changes also include contributions from other feedbacks, since their temperature responses are dependent on the temperature change from the suppressed feedbacks. In contrast the feedback parameters, λi, are invariant across the experiments given that the assumptions mentioned above hold. Later in this section it will be shown this is not fully the case when it comes to the lapse rate and surface albedo feedbacks.

The forcing F can be estimated using a data-ocean version of the model following Hansen et al. (2005). In the DOM version the SSTs and the sea ice are fixed to a monthly climatology. The forcing can be found as

 
formula

where ΔSD and ΔLD are the change in net incoming shortwave radiation and outgoing longwave radiation, respectively, at TOA associated with a doubling of CO2 in the DOM version of the model, and ΔTD is the surface air temperature change in that model. The temperature change is small and due to the fixed SSTs and sea ice it appears mostly over land. The forcing defined this way includes the radiative effect of the fast adjustments (Hansen et al. 2005). These adjustments include cooling of the stratosphere and fast cloud changes, which are mostly radiation-induced and mostly independent of the surface air temperature response. Likewise for the slab-ocean version it can be taken into account that an equilibrium state has not been fully reached by modifying Eq. (7):

 
formula

so that the changes of downward shortwave and upward longwave radiation are encountered. Equations (9) and (10) can be solved for F and λ.

Further, since the lapse rate and the surface albedo feedbacks are examined here, and since these are assumed to separately impact the longwave and the shortwave radiation, it is convenient to divide the feedback parameter into a longwave and a shortwave part (λ = λL + λS), where the lapse rate feedback is included in the former and the surface albedo feedback in the latter. Following Winton (2006), Eqs. (9) and (10) can simply be split into

 
formula

and

 
formula

where FL and FS are the longwave and shortwave component of the forcing, respectively.

Table 1 provides forcings and feedback parameters for five experiments where the lapse rate and the surface albedo feedbacks are locked in different combinations. From these experiments, estimates of the lapse rate and the surface albedo feedback parameters, λlr and λa, respectively, are obtained from

 
formula

For example, the lapse rate feedback parameter associated with suppressing the feedback using the SAT-based method, λlr(SAT), can be estimated by taking the difference between the longwave feedback parameter in the experiments including this feedback, , and the experiment where it has been locked, . If the difference is estimated on the basis of the free experiment and the locked lapse rate (SAT based) experiment, it provides the lapse rate feedback parameter in the free experiment. Likewise this lapse rate feedback parameter can be estimated in the locked surface albedo experiment by comparing this experiment with that named locked lapse rate (SAT based) and surface albedo. In a similar way the surface albedo feedback parameter can be achieved for the free and the locked lapse rate (SAT based) experiments from shortwave feedback parameters.

Table 1.

Forcings and feedback parameters for experiments where the lapse rate and the surface albedo feedbacks are locked in different combinations. The total forcing is indicated by F, and its split into longwave and shortwave parts by FL and FS. The longwave and shortwave feedback parameters are given by λL and λS. The lapse rate feedback (SAT based), λlr(SAT), is found from the difference between λL in experiments where this feedback is included and experiments where it is suppressed. A similar procedure is used to estimate λlr(TrMT), λa, , , and . The latter three indicate the change in the shortwave feedback parameter associated with the lapse rate feedback, and the change of the longwave feedback parameter associated with the surface albedo feedback. The forcings and the feedback parameters are estimated using Eqs. (11)(13).

Forcings and feedback parameters for experiments where the lapse rate and the surface albedo feedbacks are locked in different combinations. The total forcing is indicated by F, and its split into longwave and shortwave parts by FL and FS. The longwave and shortwave feedback parameters are given by λL and λS. The lapse rate feedback (SAT based), λlr(SAT), is found from the difference between λL in experiments where this feedback is included and experiments where it is suppressed. A similar procedure is used to estimate λlr(TrMT), λa, , , and . The latter three indicate the change in the shortwave feedback parameter associated with the lapse rate feedback, and the change of the longwave feedback parameter associated with the surface albedo feedback. The forcings and the feedback parameters are estimated using Eqs. (11)–(13).
Forcings and feedback parameters for experiments where the lapse rate and the surface albedo feedbacks are locked in different combinations. The total forcing is indicated by F, and its split into longwave and shortwave parts by FL and FS. The longwave and shortwave feedback parameters are given by λL and λS. The lapse rate feedback (SAT based), λlr(SAT), is found from the difference between λL in experiments where this feedback is included and experiments where it is suppressed. A similar procedure is used to estimate λlr(TrMT), λa, , , and . The latter three indicate the change in the shortwave feedback parameter associated with the lapse rate feedback, and the change of the longwave feedback parameter associated with the surface albedo feedback. The forcings and the feedback parameters are estimated using Eqs. (11)–(13).

In the free experiment the SAT-based lapse rate feedback is around −0.2 W m−2 K−1and the surface albedo feedback around 0.6 W m−2 K−1. Bitz et al. (2012) found numbers of around −0.1 and 0.3 W m−2 K−1, respectively, for the same model but with a 1° horizontal resolution and using the kernel method. The discrepancies may partly be associated with interactions between feedbacks in the sense that Eq. (5) is not fulfilled. This means that a fraction of the temperature response associated with one feedback is dependent on another feedback, which implies that the feedback parameters are dependent on one another. Such interactions would be included in the estimates of our study but are not taken into account by the kernel method. The discrepancies of the results may also be related to uncertainties of the kernel method. In particular, the surface albedo kernel has been shown to be dependent on the climate (Block and Mauritsen 2013).

Interactions between the lapse rate and the surface albedo feedback are evident: The SAT-based lapse rate feedback parameter is around −0.2 W m−2 K−1 in the free experiment, but around −0.5 W m−2 K−1 in the model version without a surface albedo feedback. Further, the surface albedo feedback is weaker in the experiment where the lapse rate is locked. This indicates that the assumption of feedbacks acting independently is questionable when it comes to the lapse rate and surface albedo feedbacks. Globally the lapse rate feedback is negative and the surface albedo feedback is positive, but in the polar areas they are both positive, which leads to interactions between the feedbacks. For instance, the surface albedo feedback induces large surface warming at high latitudes, which enhances the lapse rate feedback in this region, but has a smaller effect on the lower latitudes. As a result the structure of the lapse rate feedback is to some extent dependent on the albedo feedback.

Figure 8 shows profiles of temperature changes similar to those in Fig. 4, but for experiments with the surface albedo feedback suppressed. A comparison with the profiles from the experiment including this feedback (Figs. 4a–c) indicates that the difference between the locked and the free profiles in the polar areas is much smaller when the surface albedo is locked than when it is included. Hence the lapse rate feedback is more efficient at high latitudes when the surface albedo feedback is active. In contrast, at the equatorial latitudes the profiles are only little affected by the surface albedo feedback. As a result, the lapse rate feedback is less negative when the surface albedo feedback is active since the surface albedo feedback enhances the regionally positive lapse rate feedback at high latitudes, while it has little effect at low latitudes.

Fig. 8.

As in Figs. 4a–c, but for the experiments with suppressed surface albedo feedback. Hence “free” refers here to the locked SA experiment with suppressed surface albedo feedback, and “locked” to the locked LR (SAT) and SA experiment with both surface albedo and lapse rate feedback suppressed.

Fig. 8.

As in Figs. 4a–c, but for the experiments with suppressed surface albedo feedback. Hence “free” refers here to the locked SA experiment with suppressed surface albedo feedback, and “locked” to the locked LR (SAT) and SA experiment with both surface albedo and lapse rate feedback suppressed.

Meridional sections of longwave and shortwave feedback parameters are shown in Figs. 9a and 9b and of lapse rate and surface albedo feedback parameters by solid lines in Figs. 9c and 9d. The sections are based on zonal-mean estimates of radiation changes and global-mean estimates of the surface air temperature change, and can be derived from Eqs. (11) and (12):

 
formula

where ϕ is latitude and ˜ indicates zonal estimates. By decomposing the feedback parameters in this way, an integration over all latitudes results in the global estimates presented in Table 1.

Fig. 9.

Zonal mean of (a)–(d) feedback parameters and (e) cloud water path changes as a function of latitude. Longwave feedback parameters are displayed in (a), and those of shortwave in (b) for the free experiment with all feedbacks active, the SAT-based locked lapse rate (LR), the TrMT-based locked LR, the locked surface albedo (SA), and the experiment with both SA and SAT-based LR locked. The SAT-based lapse rate feedback parameter is displayed in (c) and the surface albedo feedback parameter in (d) by solid lines. See Table 1 for a detailed description of the feedback parameters shown in (c) and (d). Note that (c) and (d) include two estimates of either feedback. According to the linear theory the two estimates should be equal. Differences significant on a 95% level between the estimates are indicated by solid red lines at the bottom of the frames. Corresponding longwave effects of the surface albedo feedback and shortwave effects of the SAT-based lapse rate feedback are shown by dotted lines in (c) and (d), respectively. Likewise, differences significant on the 95% level between the estimates are indicated by dotted lines at the bottom of the frames. The cloud water path change shown in (e) includes liquid and frozen water.

Fig. 9.

Zonal mean of (a)–(d) feedback parameters and (e) cloud water path changes as a function of latitude. Longwave feedback parameters are displayed in (a), and those of shortwave in (b) for the free experiment with all feedbacks active, the SAT-based locked lapse rate (LR), the TrMT-based locked LR, the locked surface albedo (SA), and the experiment with both SA and SAT-based LR locked. The SAT-based lapse rate feedback parameter is displayed in (c) and the surface albedo feedback parameter in (d) by solid lines. See Table 1 for a detailed description of the feedback parameters shown in (c) and (d). Note that (c) and (d) include two estimates of either feedback. According to the linear theory the two estimates should be equal. Differences significant on a 95% level between the estimates are indicated by solid red lines at the bottom of the frames. Corresponding longwave effects of the surface albedo feedback and shortwave effects of the SAT-based lapse rate feedback are shown by dotted lines in (c) and (d), respectively. Likewise, differences significant on the 95% level between the estimates are indicated by dotted lines at the bottom of the frames. The cloud water path change shown in (e) includes liquid and frozen water.

The solid lines in Fig. 9c provide the SAT-based lapse rate feedback as a function of latitude. The feedback is positive at high but negative at low latitudes. Consistent with the discussion above, the lapse rate feedback is more positive at high latitudes when the surface albedo feedback is active than when it is suppressed.

The surface albedo feedback (Fig. 9d, solid lines) is strongest in areas where sea ice is vulnerable to melt as the global temperature rises. In addition, the surface albedo feedback is stronger when the lapse rate feedback is active, especially in the Arctic. The lapse rate feedback induces warming at the high latitudes due to the predominantly stable stratification conditions prevailing here, which hampers vertical mixing. This warming further melts the ice, thereby amplifying the surface albedo feedback.

The surface albedo feedback has a relatively strong influence on the longwave part of the feedback parameter, both for the free and for the locked lapse rate experiment. Estimates of the effect on longwave radiation of the albedo feedback, and on the shortwave radiation of the lapse rate feedback, are given as additional effects in Table 1 and with dashed lines in Figs. 9c and 9d. Since the surface albedo feedback in itself affects only the shortwave feedback parameter, the impact on the longwave part is associated with the surface albedo feedback interacting with other feedbacks. As shown here, the longwave part of the surface albedo feedback is 0.15 W m−2 K−1 when the lapse rate feedback is active, but −0.15 W m−2 K−1 when it is suppressed. Again, the difference appears mostly at the high latitudes (Fig. 9c). This is due to the longwave cooling to space in these areas being more efficient when it is taken from the locked profiles than when it is based on the profiles estimated by the model.

The SAT-based lapse rate feedback has a shortwave component of the same size as its longwave part (Table 1). This is partly linked to the lapse rate feedback interacting with the surface albedo feedback, which can be seen from a comparison between the number taken from the free and the locked surface albedo experiments. The warming at high latitudes associated with the lapse rate feedback enhances the surface albedo feedback, which results in a positive contribution of the shortwave part of the lapse rate feedback. However, the surface albedo feedback does not alone account for this shortwave part. Clouds may also play a role. In the tropics, the two experiments with suppressed SAT-based lapse rate feedback show a larger cloud increase than those where the feedback is active, which can be seen from estimates of the 2×CO2 changes of the cloud water path from each experiment (Fig. 9e). This difference in cloud water path change indicates a cloud shortwave warming effect associated with the SAT-based lapse rate feedback, which appears in the shortwave feedback parameter of this feedback. This is due to the clouds reflecting less sunlight when the lapse rate feedback is active than when it is suppressed.

As mentioned above, the longwave part of SAT-based lapse rate feedback is offset by the shortwave part in the free experiment. This is consistent with the lapse rate feedback providing only a negligible contribution to the global temperature response (Fig. 7). The direct effect of the lapse rate feedback in terms of global cooling is compensated by the impact of the lapse rate feedback on the surface albedo feedback and, presumably, on the cloud feedback. When the effect on the surface albedo feedback is suppressed, the lapse rate feedback leads to a global cooling, which can be seen from a comparison of the dark and light blue columns for the global estimates in Fig. 7.

The TrMT-based lapse rate feedback is near neutral (Table 1). With this design the feedback has little impact on the longwave energy change at TOA (Fig. 5); rather, this feedback redistributes energy between the atmosphere and the surface.

7. Conclusions

Both the lapse rate and the surface albedo feedbacks induce polar temperature amplification. Around 40% of the amplification in the Arctic and 65% of the Antarctic amplification can be attributed to the surface albedo feedback, while the SAT-based lapse rate feedback accounts for 15% of the Arctic and 30% of the Antarctic amplification.

Stable stratification conditions at high latitudes, and an increase in upper tropospheric latent heat release at low latitudes, induce differences in warming with height in the troposphere. This warming structure is associated with more radiation to space at low and less radiation to space at high latitudes, as compared to the radiation from a height-independent warming in the troposphere that is equal to that at the surface (Fig. 5a). As a result, the SAT-based lapse rate feedback is negative at low latitudes but positive at high latitudes, which leads to polar amplification. In total, the feedback is negative with a feedback parameter of ~−0.2 W m−2 K−1. This number is less negative compared to estimates obtained using the kernel method on model results from phase 3 of the Coupled Model Intercomparison Project (CMIP3) used for the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4). The kernel estimates from these models range from −0.3 to −1.3 W m−2 K−1 (Soden et al. 2008). However, our number is more negative compared to a kernel method estimate of ~−0.1 W m−2 K−1 based on the same model version, CCSM4, as used here (Bitz et al. 2012).

In comparison with the SAT-based lapse rate feedback, the feedback based on TrMT is weak in terms of inducing radiative TOA imbalance (Fig. 5b). However, when it comes to polar amplification the two ways of regarding the feedback lead to about the same results. For the SAT-based feedback, the polar amplification is due to TOA radiation changes characterized by radiative cooling to space at low and radiative warming at high latitudes, whereas for the TrMT-based feedback, it is rather a redistribution of energy between the troposphere and the surface that induces the amplification.

The surface albedo feedback parameter of ~0.6 W m−2 K−1 is at the upper end relative to estimates based on the kernel method and the CMIP3 models, which range from 0.0 to 0.5 W m−2 K−1 (Soden et al. 2008). The contribution of this feedback to the polar amplification of 40% for the Arctic and of 65% for Antarctica is in agreement with an earlier estimate by Hall (2004), but around double the amounts relative to values found by Graversen and Wang (2009), using similar methods of online suppressing the feedback.

The estimate from Graversen and Wang (2009) is based on an earlier version of the CCSM model, version 3 (CCSM3). The sea ice model component has become more sophisticated in the CCSM4 relative to the earlier version, especially when it comes to shortwave absorption (Holland et al. 2012). In CCSM4 the absorption and hence the albedo is based on inherent optical properties of the different snow and ice layers (Briegleb and Light 2007), whereas in the earlier model version the albedo is parameterized on the basis of bulk sea ice properties such as snow and ice thicknesses and surface temperature. In addition, the new version includes melt ponds, which contribute to a positive albedo feedback (Holland et al. 2012). A comparison to observations reveals that CCSM3 as well as most of the other models included in the CMIP3 archive underestimate the decline of the Arctic sea ice extent during the last decades (Stroeve et al. 2007). Furthermore, it has been pointed out that CCSM4 better represents the twentieth-century Arctic climate than does CCSM3 (Jahn et al. 2012). All in all, based on the difference in sophistication and in performance, it is likely that the CCSM4 provides a more realistic picture of the surface albedo feedback than that obtained by CCSM3.

The lapse rate and the surface albedo feedbacks strongly interact with each other and violate the linear forcing response approximation, where feedbacks are assumed independent. The lapse rate feedback is considerably stronger (more negative) in a world without the surface albedo feedback. This is due to the surface temperature amplification being smaller when the surface albedo feedback is not active, which reduces the positive lapse rate feedback at high latitudes. Another example is found in the experiment where all feedbacks are included. Here the global surface air temperature response to the lapse rate feedback is small despite the fact that the feedback parameter is negative by ~−0.2 W m−2 K−1 . This is due to the shortwave effect of the lapse rate feedback offsetting the longwave part. The shortwave effect is partly a result of the high-latitude surface warming induced by the lapse rate feedback, which further melts the sea ice whereby the surface albedo feedback is enhanced. Also a part of the shortwave effect of the lapse rate feedback seems linked to cloud changes. A shortcoming of this study is that it is based on one climate model only. A comparison with similar results from other climate models would be interesting.

The model with both the lapse rate and the surface albedo feedback suppressed shows polar amplification, especially in the Arctic. Around 40% of the amplification in the Arctic appears to be unexplained by these two feedbacks. In the absence of the surface albedo feedback, the retreat of sea ice may still lead to enhanced warming at high latitudes (Hall 2004). For instance, in the dark seasons the sea ice insulates the warm ocean from the cold atmosphere and allows the surface air temperature to become far below the freezing point. As sea ice retreats or thins, the temperature of the surface air will increase and come closer to that of the ocean.

However, when it comes to revealing the causes of the amplification, attention should also be turned toward other processes such as cloud processes, changes of the meridional energy transport, and the Planck temperature feedback.

Acknowledgments

The authors would like to thank three anonymous reviewers who provided very constructive and helpful comments. The authors are grateful for interesting and useful discussions with Rodrigo Caballero, Gunilla Svensson, Michael Tjernström, and Jonas Nycander. All experiments were run at the Triolith supercomputer from the National Supercomputer Centre (NSC), Linköping, Sweden, as projects SNIC 2013/1-101 and SNIC 2013/1-223 approved by the Swedish National Infrastructure for Computing (SNIC). The CCSM4 model was obtained from NCAR, Boulder, Colorado, United States. This work is part of the program Advanced Simulations of Climate Change and Impacts on Northern Regions (ADSIMNOR) funded by the Swedish research council FORMAS.

APPENDIX A

Feedback Parameters

Following, for example, Hansen et al. (1984), it is here shown that the two assumption given by Eqs. (4) and (5) lead to Eqs. (6) and (7). In a climate system without any feedback processes (N = 0) the temperature change ΔT0 eventually causes an offset of the radiation imbalance induced by the forcing. Let f0 be a climate sensitivity factor for such a system so that

 
formula

Dividing ΔT0 by ΔTeq, using Eqs. (4), (5), and (A1) and a bit of algebra, yields

 
formula

Now define

 
formula

Then Eq. (A2) can be written as

 
formula

which combined with Eq. (4) leads to Eq. (7). Positive (negative) feedbacks tend to increase (decrease) the radiative imbalance at the climate system boundary [Eq. (8)]. Hence, in that sense, the effect of the temperature change leading to a restoring of the energy balance at the boundary may itself be regarded as a negative feedback, consistent with the negative sign of λ0 as defined by Eq. (A3). This feedback may be regarded as the Planck feedback associated with a uniform, height-independent temperature change in the troposphere.

APPENDIX B

Statistical Significance

The statistical significance of differences in Figs. 6b, 9c, and 9d is estimated based on a t test. The t test provides the statistical significance of the difference between two means (e.g., Snedecor 1956, 85–101). The significance depends on the variances of the time series on which the means are based. For the SOM experiments, the means are taken over 80 years and for the DOM experiments over 50 years. To estimate the variances the SOM time series are broken into four consecutive 20-yr averages and the DOM time series into four 10-yr averages. The variances are taken over the four averages. It is here assumed that the consecutive averages are independent, which may not be entirely the case.

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