Previous studies show that the evolution of global mean temperature forced by the total forcing is almost the same as the sum of individual orbital, ice sheet, greenhouse gas and meltwater single forcing runs in the last 12 000 years in three independent climate models: Community Climate System Model 3 (CCSM3), Fast Met Office/UK Universities Simulator (FAMOUS) and Loch-Vecode-Ecbilt-Clio-Agism Model (LOVECLIM). This validity of the linear response is useful because it simplifies the interpretation of the climate evolution. However, it has remained unclear if this linear response is valid on other spatial and temporal scales and, if valid, in what regions. Here, using a set of TraCE-21ka (Simulation of the Transient Climate of the Last 21,000 Years) climate simulations, the spatial and temporal dependence of the linear response of the surface temperature evolution in the Holocene is assessed approximately using the correlation coefficient and a linear error index. The results show that the response of global mean temperature is almost linear on orbital, millennial and centennial scales in the Holocene but not on a decadal scale. The linear response differs significantly between the Northern Hemisphere (NH) and Southern Hemisphere (SH). In the NH, the response is almost linear on a millennial scale, while in the SH the response is almost linear on an orbital scale. Furthermore, at regional scales, the linear responses differ substantially between the orbital, millennial, centennial and decadal timescales. On an orbital scale, the linear response is dominant for most regions, even in a small area of a midsize country like Germany. On a millennial scale, the response is still approximately linear in the NH over many regions. Relatively, the linear response is degenerated somewhat over most regions in the SH. On the centennial and decadal timescales, the response is no longer linear in almost all the regions. The regions where the response is linear on the millennial scale are mostly consistent with those on the orbital scale, notably western Eurasian, North Africa, subtropical North Pacific, the tropical Atlantic and the Indian Ocean, likely causing a large signal-to-noise ratio over these regions. This finding will be helpful for improving our understanding of the regional climate response to various climate forcing factors in the Holocene, especially on orbital and millennial scales.
Long-term temperature evolution in the Pleistocene is often believed to be, and
therefore interpreted as being, driven mainly by several external forcing
factors, notably, orbital forcing, greenhouse gases (GHGs), continental ice sheets and meltwater flux forcing. (Here, we treat the
coupled ocean–atmosphere system as our climate system, such that Earth
orbital parameters, GHGs, meltwater discharge and continental ice sheet are
regarded as external forcing.) Implicit in this interpretation is often an
assumption that the response is almost linear to the four forcing factors,
that is, the temperature evolution forced by the total forcing combined is
approximately the same as the sum of the temperature responses forced
individually by the four forcing factors. This linear response, if valid,
simplifies the interpretation of the climate evolution dramatically because
each feature of the climate evolution can now be attributed to those on
different forcing factors. One example is the global mean temperature
evolution of the last 21 000 years (COHMAP members, 1988; Liu et al., 2014).
It has been shown that the global mean temperature response is almost linear
to the four forcing factors above in three independent climate models
(Community Climate System Model 3 (CCSM3), Fast Met Office/UK Universities Simulator (FAMOUS) and Loch-Vecode-Ecbilt-Clio-Agism Model (LOVECLIM); Fig. 2 of Liu et al., 2014) with the temperature
evolution forced by the total forcing almost the same as the sum of those
individually forced by each forcing factor. Furthermore, this deglacial
warming response is forced predominantly by the increase in GHGs, with
significant contribution from the ice sheet retreat. This linear response,
however, has not been assessed quantitatively for the climate evolution in
the Holocene. The Holocene period poses a more stringent and interesting
test of the linear response, as it removes the deglacial global warming
response that is dominated by that to increased
In general, the assessment of the linear response, in principle, can be done in a climate model using a set of experiments that are forced by the combined forcing as well as each individual forcing. Furthermore, each forcing experiment has to consist of a large number of ensemble members. This follows because a single realization of a coupled ocean–atmosphere model could contain strong internal climate variability on a wide range of timescales (Laepple and Huybers, 2014), from daily variability of synoptic weather storms (Hasselmann, 1976) to interannual variability of El Niño (Cobb et al., 2013) and interdecadal climate variability (Delworth and Mann, 2000), all the way to millennial climate variability (Bond et al., 1997). The ensemble mean is therefore necessary to suppress internal variability and then generate the truly forced response to each forcing. The goodness of the linear response can therefore be assessed by comparing the response to the total forcing with the sum of the individual responses. One practical problem with this ensemble approach is, however, the extraordinary computing costs, especially for long experiments in more realistic, fully coupled general circulation models. A more practical question is therefore: is it possible to obtain a meaningful assessment of the linear response using only a single realization of each forced experiment for the Holocene, such as those in TraCE-21ka (Simulation of the Transient Climate of the Last 21,000 Years) experiments (Liu et al., 2014).
Strictly speaking, it is impossible to disentangle the forced response from internal variability in a single realization. This would make the assessment of the linear response difficult. However, it is conceivable that, if our interest is the slow climate evolution of millennial or longer timescales in response to the slow forcing factors such as the orbital forcing, ice sheet forcing, GHGs and meltwater flux, the assessment is still possible, albeit approximately, at least for very large-scale variability. This follows because these forcing factors are of long timescales and of large spatial scales; the forced response signal should therefore also be on long timescales and large spatial scales if the response is approximately linear. An extreme example is the almost linear response in the global temperature of the last 21 000 years as discussed by Liu et al. (2014). In contrast, internal variability in the coupled ocean–atmosphere system tends to be of shorter timescales (decadal to centennial) and of smaller spatial scales, at least in the current generation of coupled ocean–atmosphere models. This naturally leads to two questions. First, how linear is the climate response at different spatial and temporal scales, quantitatively? Second, in what regions does the linear response tend to dominate? The answer to these questions should help improve our understanding of regional climate response during the Holocene. A further question is as follows: if the linear approximation is valid, what is the contribution of each forcing factor in different regions and at different timescales. This question will be addressed in a follow-up paper (Wan et al., 2019).
In this paper, we assess the linear response for the Holocene temperature evolution quantitatively, using five forced climate simulations in CCSM3 (Liu et al., 2014), with the focus on the spatial and temporal dependence of the linear response. We will assess the linearity response to orbital, millennial, centennial and decadal timescales and on global, hemispheric and regional spatial scales. The data and methodology are given in Sect. 2. The dependence of the linear response to spatial and temporal scales is analyzed in Sect. 3. A summary and further discussions are given in Sect. 4.
TraCE-21ka simulation experiments.
The data are from TraCE-21ka (Liu et al., 2009, 2014), which consists of a
set of five synchronously coupled atmosphere-ocean general circulation model
simulations for the last 21 000 years. The simulations are completed using
the CCSM3 (Community Climate System Model version 3). The simulation forced
by the total forcing (experiment ALL) is forced by realistic continental ice
sheets, the GHGs, orbital forcing and meltwater fluxes. The ice sheet is
changed approximately once every 500 years, according to the ICE-5G
reconstruction (Peltier, 2004). The atmospheric GHG concentration is
derived from the reconstruction of Joos and Spahni (2008). The orbital
forcing follows that of Berger (1978). The coastlines at the LGM (Last Glacial Maximum) were also
taken from the ICE-5G reconstruction and were modified at 13.1, 12.9,
7.6 and 6.2 ka, after which the transient simulation adopted the present-day
coastlines. The meltwater flux follows largely the reconstructed sea level
and other paleoclimate information and, in the meantime, reconciles the
response of Greenland temperature and AMOC (Atlantic Meridional Overturning Circulation) strength in comparison with
reconstructions. More information on the details of the experiment and
forcing can be seen in He (2011) or on the TraCE-21ka website
The transient simulation under the total climate forcing reproduces many large-scale features of the deglacial climate evolution consistent with the observations (Shakun et al., 2012; Marsicek et al., 2018), suggesting a potentially reasonable climate sensitivity in CCSM3, at global and continental scales. In addition to the all-forcing run (ALL), there are four individual forcing runs forced by the orbital forcing (ORB), the continental ice sheets (ICE), the GHGs (GHG) and meltwater forcing (MWF) (Liu et al., 2014; Table 1). In these four experiments, only one forcing varies the same as in experiment ALL, while other forcings/conditions remain the same as at 19 ka. Therefore, this set of experiments can be used to study the linear response of the climate to the four forcing factors. Here, we will only examine the surface temperature response in the Holocene (last 11 000 years).
We will use correlation and normalized root means square error (RMSE) to assess the linear response
(see next subsection for details). We note, however, that our assessment of
the linear response is approximate. Before introducing the details of the
assessment method, it is useful here to make some general comments on the
linear response assessment. As pointed out by one reviewer, strictly
speaking, the assessment of the linear response requires one to answer two
questions.
How linear is the response to external forcing? What is the relative importance of external forcing vs. internal
variability, assuming the response were linear?
Specifically, for Q1, if we denote the temperature response to the full
external forcing by
In spite of these potential issues, with a single member for each experiment, useful information can still be extracted on linear response. Our general hypothesis is that the slow (orbital and millennial) and large (continental and basin) variability is composed mostly of forced signal and the faster (centennial and shorter) and smaller variability is mostly associated with internal variability of noise. In other words, in our set of single member of simulations, the signal-to-noise ratio is large for slow variability but small for faster variability. Qualitatively, this hypothesis seems reasonable. First, all the four external forcing factors are of slow timescales and large spatial scales; additionally, internal variability is usually weak in the coupled ocean–atmosphere system at slow timescales and large spatial scales. Our focus is indeed the slow variability and large scale here, so we can roughly treat the slow and large variability in the single realization as the signal and the linearity of the response may be assessed using Eq. (2). Second, again, because our forcing factors are of slow timescales and large spatial scales, higher-frequency or small-scale variability in the model should not be dominated by the forced variability (unless the response is highly nonlinear). Therefore, high-frequency or small-scale variability can be treated roughly as “noise”. This is consistent with the later assessment that slow variability seems to be an approximately linear response while high-frequency variability not. Based on this hypothesis, the signal-to-noise ratio is also estimated using the variance of slow variability as the signal and high-frequency variability as the noise (as in late Fig. 7). It should be noted however that this hypothesis is qualitative in nature. One major purpose of this paper is to give a somewhat more quantitative assessment on this hypothesis. How slow, how large and how good will the linear response be?
Our experimental design is proper for linear response assessment here. Alternatively, in another experimental setting, individual forcing experiments are often superimposed sequentially one by one: for example, first the ice sheet, second the ice sheet plus orbital forcing, third the ice sheet, orbital and GHGs, and finally, applying all four forcings of ice sheet, orbital, GHGs and meltwater. In this experimental design, the full forcing response is by default the response of the sum response after adding the four forcing factors together, and therefore it cannot be used to assess the linearity of the response. Nevertheless, it should be kept in mind that our four individual forcing experiments are not designed optimally for the study of the linear response in the Holocene. This is because, except for the variable forcing, all the other three forcing factors are fixed at the 19 ka condition. As such, the mean state is perturbed from the glacial state, instead of a Holocene state. This may have contributed to some unknown deterioration on the linear response discussed later. Nevertheless, we believe, our major conclusion should hold reasonably well. This is because, partly, the response is indeed almost linear for orbital and millennial variability as will be shown later.
We use two indices to evaluate the linear response: the temporal correlation
coefficient
But how to assess the goodness of the linear response from the value of
The statistical significance of
The statistical significance of the
The dependence of the linear response on spatial and temporal scales will be
studied by filtering the time series on different scales. For the spatial
scale, we will divide the globe into nine successive cases, denoted by nine division factors:
On the timescale, we decompose a full 11 000-year annual temperature time
series (from 11 to 0 ka) in 100-year bins (a total of 110 data bins or
points, each representing a 100-year mean) into three components. The three
components are to represent the variability of, roughly, orbital, millennial
and centennial timescales. Following Marsicek et al. (2018), we derive the
orbital and millennial variability using a low-pass filter called the
locally weighted regression fits (Loess fits) (Cleveland, 1979). First, the
orbital variability is derived by applying a 6500-year Loess fit low-pass
filter to the temperature time series, and it therefore contains the trend
and the slow evolution longer than
Given the different degrees of freedom especially among the filtered
variability of different timescales, it is important to test the goodness
of the linear response statistically on
The global annual mean surface temperature time series derived
from the ALL run (black) and the SUM (the sum of four single forcing runs,
red). In
In this paper, this AR(1) test for global mean temperature is also used as the common significant test for different spatial scales and in different regions as well. This use of a common significance level is for simplicity here. First, the use of different regional AR(1) coefficients for different regions will make the comparison of the linear responses among different spatial scales (e.g., Figs. 3 and 4) and different regions (Figs. 5 and 6) difficult. Second, except for the orbital timescale, the AR(1) coefficient for the global mean temperature is larger than most of the regional AR(1) coefficients (not shown), likely caused by the further suppression of internal variability in the global mean. As a result, the global mean AR(1) test actually serves as a more stringent test than the local AR(1) test. At the orbital scale, the global mean AR(1) coefficient is in about the middle of the regional AR(1) coefficients. The uncertainly of using the global mean AR(1) coefficient is therefore about the average of those of regional AR(1) coefficients. Third, and, most importantly, as our first study here, our focus is on the global features of the linear response. The difference among the AR(1) coefficients among different spatial scales and different regions is much smaller than that between different timescales here. Therefore, the global mean AR(1) can still provide an approximate guideline for the proper significant test at different timescales. In later studies, if one's focus is on a specific spatial scale and on a specific region, the regional AR(1) should be used to reexamine the significance test.
As a reference, the significance level of
The global mean temperature provides a useful example to start the
discussion of the dependence of the linear response on timescales. We first
examine the linear response of the global mean temperature based on its
components of orbital, millennial, centennial and decadal variability
(Fig. 1). Figure 1a is the total variability of global surface temperature
derived from the ALL run and the sum of the four individual forcing
experiments. The global temperature response is almost linear on the orbital
and millennial scales throughout the Holocene (Fig. 1b and c). The orbital-scale evolution is characterized by a warming trend of about 1
The surface temperature time series derived from the ALL run
(black) and the SUM (red). Panels
In order to assess the linear response at different spatial scales, we first
analyze the linear response to the hemisphere scale for the NH and SH
(
The linear response at different spatial scales and on the orbital,
millennial, centennial and decadal timescales is summarized in Figs. 3 and 4 in the correlation coefficient and linear error index, respectively.
Figure 3a shows the correlation coefficients of the orbital variability in each
region for the nine division factors. The cases of global mean (
Millennial variability also shows a weaker linear response for smaller
scales, in both the correlation (Fig. 3b) and linear error (Fig. 4b).
Quantitatively, for millennial variability, the response is still
approximately linear in the NH over many regions, albeit less so than at the
orbital scale. The correlation coefficients remain above 0.6 across most
regions even at the smallest division area (
In contrast to orbital and millennial variability, almost no response can be
confirmed as linear for centennial variability. The median linear response is
no longer significant on the centennial timescale in either hemisphere
across spatial scales (
Correlation coefficient of mean surface temperature between the
ALL and SUM outputs in different spatial timescales.
The approximate linear response at the orbital and millennial scales suggests that these two groups of variability are generated predominantly by the external forcing. In contrast, the poor linear response of centennial and decadal variability suggests that these two groups of variability are caused mainly by the internal coupled ocean–atmosphere processes. This is largely consistent with our original hypothesis. It should be kept in mind that, in our single realization here, the poor linear response to centennial and decadal variability may also be contributed by nonlinear responses of the climate system. But, given the almost complete absence of forcing variability at this short timescale in our experiments, we do not think that the nonlinear response is the major cause of the poor linear response here.
Same as Fig. 3 but for linear error (
We now further study the pattern of the linear response. Figure 5 shows the
spatial patterns of the correlation coefficients at orbital (Fig. 5a1–a3) and
millennial (Fig. 5b1–b3) scales for three representative spatial scales:
Correlation coefficient of mean surface temperature between the
ALL and SUM outputs of the two timescales (
For millennial variability, in the NH, the linear response shows a similar
feature to that of orbital variability, but the linear response is poor over
almost all the SH. Figure 5b1–b3 show that the response is almost linear in
most regions in the NH at the three spatial scales, with the correlation
coefficient above 0.6. At the regional scale, e.g.,
The signal-to-noise ratios on the orbital
In order to understand the cause of the preferred regions of the linear
response, we examine the signal-to-noise ratio. As discussed in Sect. 2.2, our forcing factors are on millennial and orbital timescales, and the
linear response is also largely valid for orbital and millennial
variability. We will therefore use the variance of the orbital and
millennial variability as a crude estimate of the linear response signal.
Similarly, since there is no centennial and decadal forcing in our model and
the response of centennial and decadal variability is not a linear response,
we use the variance of the sum of the centennial and decadal variability as
a rough estimate for internal variability as the linear noise. Admittedly,
this estimation is crude, limited by the single realization here. This
signal-to-noise ratio does not directly address Q2 in Sect. 2.2 because the timescales of the signal and noise are different. Instead, it
is used as a rough estimation of the relative magnitude of the signal-to-noise
ratio between different regions, with the assumption that the relative noise
level between different regions may be not too sensitive to the timescales.
Indeed, the use of the signal-to-noise ratio here is to shed some light on the
regional preference of the linear response. Figure 6 shows the signal-to-noise
ratio for orbital and millennial variability for the three representative
spatial scales (
Scatter diagram of the correlation coefficient and the
signal-to-noise ratio (variance ratio) on the orbital
For millennial variability, the signal-to-noise ratio also shows a similar
feature to that of orbital variability although overall somewhat smaller
(note the different color scales). Figure 6b1–b3 show that the signal-to-noise
ratio is large in most regions in the NH in all three spatial scales, with
the signal-to-noise ratio above 10
In this paper, the linear response is assessed for the surface temperature response to orbital forcing, GHGs, meltwater discharge and continental ice sheet throughout the Holocene in a coupled GCM (general circulation model; CCSM3). The global mean temperature response is almost linear on the orbital, millennial and even centennial scales throughout the Holocene but not for decadal variability (Fig. 1). Furthermore, the sum response accounts for over 50 % of the total response variance for orbital and millennial variability. Further analysis on the regional scale suggests that the response is approximately linear on the orbital and millennial scales for most continental regions over the NH and SH, with the sum response explaining over about 50 % of the total response variance. However, the linear response is not significant over much of the ocean, especially over the ocean in the SH. There are specific regions where the linear response tends to be dominant, notably the western Eurasian continent, North Africa, central and South America, the Antarctic continent, and the North Pacific. The strong linear response is interpreted as the region of large signal-to-noise ratio. That is, in these regions, either the orbital and millennial response signal is large or the influence of the centennial and decadal variability noise is small or both. This suggests that the orbital and millennial variability in these regions is relatively easy to understand. This finding lays a foundation for our further understanding of the impacts of different climate forcing factors on the temperature evolution in the Holocene of orbital and millennial timescales. This understanding is our original motivation for this work. Further work is underway in understanding the contribution of different forcing factors on the temperature evolution (Wan et al., 2019).
It should be kept in mind that since there is only one member for each experiment, we cannot separate the forced response signal from the internal variability of noise clearly at each timescale. Therefore, we cannot address Q1 and Q2 raised in Sect. 2.2 accurately. Instead, our assessment is likely contaminated by internal variability (see discussions in Sects. 1 and 2). In particular, for smaller-scale variability, of which internal variability is likely to be strong and the forced signal is likely to be weak, our correlation may underestimate the linearity of the response (see Eq. 2). Nevertheless, we speculate that our results on large-scale variability still remain robust. Furthermore, at regional scales, although the absolute value of the linear correlation of the forced response may be underestimated, it is possible that relative between different regions, the linear assessment may still be somewhat valid. These speculations, however, require much further study, especially with ensemble experiments. In spite of its limitation, our study represents the first systematic assessment of the linear response for the Holocene and can serve as a starting point for further studies in the future.
There are many further issues that need to be studied. Our study here is carried out for a single variable (surface temperature) in a single model (CCSM3) for the Holocene. Yet, the linear response could differ for different variables, in different models, for different periods and for different sets of forcing factors. For example, if we evaluate the precipitation response in the Holocene in CCSM3, the response is less linear than temperature (not shown); this is expected because the precipitation response contains more internal variability and exhibits more nonlinear behavior than temperature. The assessment will be also different if a different period is assessed, e.g., the last 21 000 years; with a large amplitude of climate forcing, the linear response may degenerate in the 21 000-year period. In addition, the assessment of the linear response using only one realization will be difficult to perform for volcanic forcing and solar variability forcing; these forcing factors have short timescales and therefore their impacts will be difficult to separate from internal variability without ensemble experiments. Finally, it is also important to repeat the same assessment here in different models and to establish the robustness of the assessment. It should also be kept in mind that our assessment is implicitly related to the assumption that, at millennial and orbital timescales, internal variability is not strong relative to the forced responses. Although this seems to be consistent in our model, there is a possibility that internal variability is severely underestimated in the model compared to the real world (Laepple and Huybers, 2014). If true, the relevance of our model assessment to the real world will be limited. It should also be kept in mind that, if the response is dominated by that to a single forcing, the assessment of the linear response here becomes one that is more relevant to the question of the forced response vs. internal variability, as discussed in Q2 in Sect. 2.2. As a further step, though, one can examine if the magnitude of the total response responds to the magnitude of this single forcing linearly.
Even in the context of this model assessment, much further work remains. Most importantly, the purpose of testing the linear response is for a better understanding of the physical mechanism of the climate response. It is highly desirable to understand why the response tends to be linear in some regions but not in others. In particular, it is unclear why the linear response is preferred over land than over ocean for orbital and millennial variability. At such a long timescale, one would expect that the upper-ocean response has reached quasi-equilibrium and therefore the surface temperature response over land and over ocean should not be too different. Ultimately, we would like to assess and understand the physical mechanism of the climate evolution in different regions. This work is underway (Wan et al., 2019).
The TraCE-21ka data sets were provided by NCAR from their website at
LW, JL and ZL conceived the idea for the paper; LW carried out the analysis and prepared the first draft. ZL contributed great ideas and gave the suggestion for the analysis. JL and ZL revised the paper several times. WS provided some useful suggestions to the paper. BL helped to download the larger TraCE-21ka data. All coauthors helped to improve the paper.
The authors declare that they have no conflict of interest.
The authors thank Kai Ding for checking the grammar of the first draft. The TraCE-21ka data sets were provided by NCAR.
This work was jointly supported by the National Key Research and Development Program of China (grant no. 2016YFA0600401), the National Natural Science Foundation of China (grant no. 41420104002, 41630527), the Program of Innovative Research Team of Jiangsu Higher Education Institutions of China and the Priority Academic Program Development of Jiangsu Higher Education Institutions (grant no. 164320H116), and NSF OCN1810681.
This paper was edited by Qiuzhen Yin and reviewed by Oliver Bothe and two anonymous referees.