ESDEarth System DynamicsESDEarth Syst. Dynam.2190-4987Copernicus PublicationsGöttingen, Germany10.5194/esd-7-597-2016Comment on “Scaling regimes and linear/nonlinear responses of last millennium
climate to volcanic and solar forcing” by S. Lovejoy and C. Varotsos (2016)RypdalKristofferkristoffer.rypdal@uit.noRypdalMartinDepartment of Mathematics and Statistics, UiT The Arctic University of Norway, Tromsø, NorwayKristoffer Rypdal (kristoffer.rypdal@uit.no)13July20167359760913March201618March201625June201627June2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://esd.copernicus.org/articles/7/597/2016/esd-7-597-2016.htmlThe full text article is available as a PDF file from https://esd.copernicus.org/articles/7/597/2016/esd-7-597-2016.pdf
(L&V) analyse the temperature response to solar, volcanic, and solar plus
volcanic forcing in the Zebiak–Cane (ZC) model, and to solar and solar plus
volcanic forcing in the Goddard Institute for Space Studies (GISS) E2-R model. By using a simple wavelet filtering
technique they conclude that the responses in the ZC model combine
subadditively on timescales from 50 to 1000 years. Nonlinear response on
shorter timescales is claimed by analysis of intermittencies in the forcing
and the temperature signal for both models. The analysis of additivity in the
ZC model suffers from a confusing presentation of results based on an invalid
approximation, and from ignoring the effect of internal variability. We
present tests without this approximation which are not able to detect
nonlinearity in the response, even without accounting for internal
variability. We also demonstrate that internal variability will appear as
subadditivity if it is not accounted for. L&V's analysis of intermittencies
is based on a mathematical result stating that the intermittencies of forcing
and response are the same if the response is linear. We argue that there are
at least three different factors that may invalidate the application of this
result for these data. It is valid only for a power-law response function; it
assumes power-law scaling of structure functions of forcing as well as
temperature signal; and the internal variability, which is strong at least on
the short timescales, will exert an influence on temperature intermittence
which is independent of the forcing. We demonstrate by a synthetic example
that the differences in intermittencies observed by L&V easily can be
accounted for by these effects under the assumption of a linear response. Our
conclusion is that the analysis performed by L&V does not present valid
evidence for a detectable nonlinear response in the global temperature in
these climate models.
Introduction
The issue of linearity in the global
temperature responses of modern general circulation models (GCMs) and Earth
system models (ESMs) is important because the prospect of predicting global
aspects of the climate under different forcing scenarios is considerably
brighter if the response is reasonably linear. Linear-response models with
two characteristic response times or a long-memory power-law response have
had considerable success in describing global temperature response in GCM
data, instrumental data and in multiproxy reconstructions
.
The credibility of these results depends crucially on the validity of the
linear approximation in the global response. Particularly relevant is
, who estimate the parameters of a linear two-box energy
balance model by data from runs of a large number of CMIP5 (Coupled Model Intercomparison Project) ESMs with
step-function forcing and linearly increasing forcing. Very
good fits to the simulated global temperature are found in this study, with
the same values of the two-box model parameters for the two different forcing
scenarios. This is a very clear demonstration of the approximate linearity of
the global temperature response in the CMIP5 ensemble. The issue of
additivity of the temperature response in GCMs has been extensively studied
over the last 2 decades, and the majority of studies find only weak
nonlinearities in the global response, although nonlinearites are often found
in regional responses in some models
.
The paper by (in the following denoted L&V) is a research
paper but has the character of a review of earlier papers of Shaun Lovejoy
and coworkers. The review style has the unfortunate effect of masking the
substance of the new results presented, which is an analysis of the responses
in two different climate models to solar and volcanic forcing, and to
combinations of these forcings. The actual analysis is made in Sect. 3.4 of
the L&V paper, where the authors test the additivity of responses to solar
and volcanic forcing in the Zebiak–Cane (ZC) model, and in Sect. 4.2, where
they study the intermittency of forcing and response and conclude that
difference in their intermittency implies nonlinearity of the response. In
Sect. of this comment we present a critical examination
of the methods L&V invoke to conclude that combined solar and volcanic
forcing leads to a weaker response than the sum of the solar and volcanic
responses in the ZC model. Section examines the
intermittency analysis and demonstrates that L&V's results for the ZC model
can be reproduced in the response of a simple linear-response model. In
Sect. we discuss some aspects of the physics that may
give rise to a nonlinear response and summarise our main conclusions.
Linearity and response additivityThe logic of hypothesis testing
According to a widely accepted principle in the philosophy of science
, a well-posed scientific hypothesis has to be
falsifiable by experiment or observation. There is an infinity of
ways the temperature response can be nonlinear. This pertains to both details
of the nonlinear interactions and to their magnitude. No test is infinitely
accurate, so there will always be a possibility that a weak nonlinearity goes
undetected. Hence, it is not logically possible to formulate a falsifiable
hypothesis stating that the response is nonlinear. The well-posed hypothesis
is that the response is linear. From this hypothesis one can design tests by
which the hypothesis can be rejected by conceivable outcomes of experiments
or observations. If such a test fails to reject the linearity hypothesis, we
cannot conclude that the response is linear, but if a series of increasingly
sharper tests still fail to reject it, the linearity hypothesis will stand
stronger. This is the principle of induction. On the other hand, if a test
turns out to reject linearity, then we have detected a nonlinearity. So, even
though nonlinearity cannot be falsified, it can in fact be verified. This is
because nonlinearity is the negation of the falsifiable linearity hypothesis;
if a statement A is false, then the statement not A is true.
Based on this logic, the only reasonable approach is to formulate a test that
may, or may not, reject the hypothesis that the response is linear. The
hypothesis, however, must be formulated with some care. The issue in the L&V
paper is nonlinearity in the response of hydrodynamic flow models like the ZC
and GCMs, which are known to be inherently nonlinear. It is not difficult to
devise tests that will detect nonlinearities in these models. The question at
hand, however, is not whether nonlinearities are present but whether these
nonlinearities are detectable in the global temperature response.
In GCM-type models “unforced” control simulations are of course driven by
the constant solar energy flux, and this results in a turbulent, nonlinear
cascade that forms the “internal variability” of the model. In a linear
model for the global response this internal variability is represented as a
noise process ϵ(t) in a global variable T(t). Forcing F(t) in
the model means a variation of the global energy flux around the flux that
drives such a turbulent equilibrium state.
The linear-response hypothesis
After these remarks we are ready to formulate the
linear-response hypothesis:
For realistic strength of the global forcing the statistics of the
internal variability ϵ(t) is unaffected by the forcing.
The global temperature can be expressed as a sum of this internal
variability and a linear response to the forcing, i.e.T(t)=Tdet(t)+ϵ(t),Tdet(t)=L^[F(t)],where T(t) is the global surface temperature; Tdet(t) is the
deterministic, linear response to the global forcing F(t); and L^ is
the linear-response operator.
Internal noise and response additivity
The data used from the ZC model are the temperature (more precisely, the
Niño3 index) after averaging over 100 simulations with the same forcing .
If the internal variability is a persistent noise, averaging over N
independent runs will reduce the standard deviation by a factor
N-1/2=0.1, but the correlation structure of the noise will be preserved.
In the following, ϵ(t) is the noise that remains after averaging
the internal noise over those N realisations.
The next step is to produce a fluctuation ΔT(t,Δt) by means of
a linear low-pass filtering operation. It could for example be a simple
moving average over a window Δt, or the Haar wavelet smoothing
employed by L&V. In the following we shall for notational simplicity omit
the arguments (t,Δt). The results presented hold for the temperature
signal itself (Δt=0) as well as for any degree Δt of
filtering. Since the response operator L^ is linear, we have
ΔTv+sdet=ΔTsdet+ΔTvdet,
where ΔTsdet and ΔTvdet are the responses to the solar and volcanic
forcings, ΔFs and ΔFv, respectively, and
ΔTv+sdet is the response to the combined forcing
ΔFs+ΔFv. This yields
ΔTs=ΔTsdet+Δϵs,ΔTv=ΔTvdet+Δϵv,ΔTs+v=ΔTsdet+ΔTvdet+Δϵs+v.
Here Δϵs, Δϵv, and Δϵs+v are the filtered fluctuations of independent
realisations of the same noise process ϵ(t) (here ϵ(t) is
the average over 100 realisations of internal variability). By subtracting
Eqs. () and () from Eq. (), and using
Eq. (), we find
ΔTs+v-ΔTs-ΔTv=Δϵv+s-Δϵs-Δϵv≡Δε.
Here, Δε is the sum of three independent realisations of the
same noise process Δϵ=dΔϵs=dΔϵv=dΔϵv+s, where =d is identity in distribution.
This implies that
Δε=d3Δϵ.
Hence, a prediction based on the linear-response hypothesis is that the
difference between the temperature driven by combined solar and volcanic
forcing and the sum of the temperatures driven by solar and volcanic forcing
is the realisation of a noise process which is 3 times the internal
variability process. In Sect. we shall test this
prediction on the data from the ZC model. If the prediction is inconsistent
with the data, the linear-response hypothesis is rejected for this model, and
nonlinearity in the response has been detected. If the prediction is
confirmed by the data, the linear hypothesis stands stronger.
Alternative test of additivity in the ZC model
Figure a shows time series of the solar and volcanic forcing for
the last millennium used in the simulations of the ZC model. Unfortunately
L&V did not have available control runs on the millennial scale from this model.
This would have been very useful in establishing directly the statistical
properties of the internal noise ϵ(t). The approach we will use as
an alternative is to assume the validity of the linear-response hypothesis,
which will allow us to extract the internal noise from the simulation with
solar forcing only. Then we will formulate a test by which the hypothesis
could be rejected by the data for volcanic forcing only and for volcanic plus
solar forcing. Assuming the validity of the linear hypothesis from the start
may seem like circular reasoning, but it is not. Any valid hypothesis testing
makes predictions based on the hypothesis, which are then tested against
observation.
(a) Time series of the solar (black) and volcanic forcing
(blue) for the last millennium used in the simulations of the ZC model.
(b) Responses after averaging over 100 realisations. The thin orange
curve is response to solar forcing; the thick orange curve is filtered by a
50-year moving average. The thick black curve is the filtered and shifted
solar forcing signal ΔTsdet(t,Δt) given by
Eq. (). The thin brown curve is the internal noise ϵ(t)
defined in Eq. (), and the thick brown curve is the filtered time series.
(c) The thin blue curve represents Ts+v(t), the thin red
curve is Ts(t)+Tv(t), and the thin black curve is
their difference ε(t)=Ts(t)+Tv(t)-Tu+v. Thick curves are the corresponding filtered series.
(d) Haar structure function of ϵ(t) (orange bullets), of
ε(t) (red bullets), and of 3ϵ(t) (brown
bullets).
If the linear-response hypothesis is true, we can determine ϵ(t) from
the solar forcing signal and the corresponding temperature signal. The solar
forcing signal in Fig. a has a smooth appearance, in particular for
the first 750 years of the record, for which no sunspot counts were
available. As a contrast, the corresponding temperature signal shown as the
thin orange curve in Fig. b is noisy on all scales down to the
annual scale. This appearance of the temperature signal under the smooth
solar forcing already lends support to the assumption that the variability up
to the century timescale is internal. However, according to L&V the
subadditivity is most prominent on timescales longer than 50 years, so we
have to pay special attention to the slow components of the noise spectrum.
We now write a linear response to the solar forcing in the form
ΔTsdett,Δt=-SΔFst-τ,Δt.
Here Δt=50 years, over which we have performed a moving average of the
temperature and forcing. The time lag τ of the response is estimated to
be ≈ 25 years from inspection of the filtered time series. The
climate sensitivity S is chosen to give the best least-squares fit of
ΔTsdet(t,Δt) (the black curve in
Fig. b) to the filtered temperature signal ΔTs(t,Δt) (the thick orange curve).
Because of the smooth character of the solar forcing signal in the first
750 years of the record, the 50-year filtering of this signal has almost no
effect, and we can therefore interpret the black curve in Fig. b as
the linear, deterministic response to the solar forcing and the difference
between the thin orange curve and the black curve as the internal noise,
i.e.
ϵ(t)=Ts(t)-ΔTsdet(t,Δt).
This difference is plotted as the thin brown curve at the bottom of
Fig. b, and the thick brown curve is the 50-year moving average.
We have now distinguished the internal noise from the solar-driven
temperature signal by means of the very simple linear-response assumption,
Eq. (). This response function is of course not accurate; the delay
in the response should rather be expressed as a time-dependent response
function (a frequency-dependent transfer function) than as a fixed delay
. For the ZC model we do not have detailed information about
the response function, so we have no means of constructing one that is known
to be better than Eq. (). But for the present purpose this is not
crucial since the solar forcing has almost no power in the high frequencies.
The orange bullets in Fig. d are a characterisation of this noise by
means of the Haar structure function (SF) employed by L&V. The definition of this
structure function is
S2Haar(Δt)=〈|ΔT(t,Δt)|2〉1/2,
where 〈…〉 denotes averaging over disjoint time intervals
of length Δt. It measures the root-mean-square (rms) fluctuation level on
the scale Δt. The flat appearance on scales above a decade indicates
a strongly persistent noise process with equally strong fluctuations on
scales Δt>10 years. The straight-line character of the log–log plot
in this scale range is symptomatic of a scaling process, and the
corresponding power spectral density has the form ∼f-β, where
β≈1 (sometimes denoted 1/f noise or pink noise). The higher
fluctuations for Δt<10 years are characteristic for the El
Niño–Southern Oscillation (ENSO). This mode is particularly strong in the ZC
model, which is designed specifically for the study of ENSO, and the global
output T(t) is the so-called Niño3 index.
If the characterisation we have made of the internal noise is correct, and
the linear hypothesis is true, then Eq. () must be true. But
ε in Eq. () must be computed from Eq. (), which
requires the temperature signals Tv and Tu+v, in
addition to Ts. The characterisation of ϵ only used
Ts, so if the linear hypothesis is false, it is very unlikely
that the estimated ε and ϵ will give good agreement with
Eq. (). This means that we should have a strong test.
In Fig. c the thin blue curve represents Ts+v(t), the
thin red curve is Ts(t)+Tv(t), and the thin black
curve is their difference ε(t)=Ts(t)+Tv(t)-Tu+v. Note that the narrow spikes from the fast
responses to the volcanic eruptions are completely absent in the difference
signal ε(t), demonstrating that the addition of solar forcing
does not exert a detectable influence on the response to the volcanic
eruptions on the short timescales up to a few years. The thick curves in
Fig. c are the corresponding 50-year moving averages. The Haar
structure function of the signal ε(t) is shown as the red bullets
in Fig. d. The brown bullets are 3ϵ(t), i.e.
the orange bullets multiplied by 3. We observe that the red and
brown bullets are more or less on top of each other; the two curves are
entangled for Δt>10 years. This means that the second-order
statistics of the noise processes ε(t) and 3ϵ
are indistinguishable, in agreement with Eq. (). Thus, this test is
not able to reject the linear-response hypothesis.
Brown bullets: Haar fluctuation function of 3ϵ(t),
where ϵ(t) is the first 195 years of the volcanic forcing record.
Red bullets are Haar fluctuations of ε(t). These two curves look
similar to the corresponding curves in . The crucial issue is
whether the difference between these two curves is statistically significant.
The thin curves constitute Haar fluctuations of a 100-member ensemble of
fractional Gaussian noises (fGn's) of 195-year length with H=-0.1
(β=2H+1=0.8). On timescale less than 10 years the fGn is not a good
model for the internal noise because of the ENSO dynamics, but on longer
timescales the flat Haar fluctuation curve suggests that an fGn with
β≈0.8 is a crude statistical model of the internal variability.
The scatter of the Haar fluctuation in this ensemble gives an idea about the
statistical uncertainty of an estimate of internal variability based on a
195-year-long record. This uncertainty exceeds the estimate of
|3ϵ(t)-ε(t)| (the difference between the brown
and the red curves); hence this difference is not statistically
significant.
This test would have been stronger if we had had a more direct estimate of the
internal variability. In an interactive comment (SC3), suggest to use a
different estimate of the internal noise, namely the first 195 years of the
volcanic-driven response time series. This is justified, since there was no
volcanic forcing in this period. The drawback, however, is that an estimate
of the Haar fluctuation from such a short time series is associated with
higher estimation uncertainty (finite sample size errors). Unfortunately,
they make no attempt to demonstrate that the estimates of the difference
|3ϵ(t)-ε(t)| are significantly different from zero
in a statistical sense. Such a test is easy to make by creating a Monte Carlo
ensemble of time series containing 195 data points with statistical
properties similar to those of the observed volcano response. The statistical
scatter of the Haar fluctuations within this ensemble will give us
information about the finite sample uncertainty of the Haar estimate. This is
done in Fig. , where the specifications of the Monte Carlo ensemble
are described in the caption. The figure shows that the difference between
the Haar fluctuations of 3ϵ(t) and ε(t) is
smaller than this uncertainty in the interesting scale range Δt>10 years. This means that the deviation from linearity observed is
statistically insignificant and hence does not reject the linear-response
hypothesis. A similar Monte Carlo ensemble for 1000-year-long time series
would reduce the scatter in the Haar fluctuations by approximately a factor
195/1000≈0.44, which is still large enough to conclude that
the difference between the blue and brown bullets in Fig. d is not
statistically significant.
Examination of L&V's test of response additivity
The L&V test of additivity shown in their paper is simpler than described
in Sect. but ignores internal variability. Here we
shall demonstrate that their test also fails to reject the linearity
hypothesis, even when this variability is not taken into account. Their main
conclusion concerning additivity of responses in the ZC model is that for
Δt>50 years the rms ratio,
R≡〈|ΔTs+ΔTv|2〉〈|ΔTu+v|2〉,
is found to be R≈1.5. As will be shown below, our analysis yields a
number indistinguishable from unity. But the authors also make attempts in
their Fig. 3 to inflate this ratio further by presenting results for the
numerator based on the flawed approximation of neglecting the estimate of
〈ΔTsΔTv〉. The approximation is
flawed because, even though solar and volcanic forcing are independent
processes, the ensemble average 〈…〉 is estimated from
only one realisation of each of these forcing processes. On the short
timescales the approximation makes sense, since the ensemble average is replaced
by time averages, but as the timescales Δt approaches the length of
the time series, the number of independent time windows to average over goes
to zero. In their Fig. 3b L&V show the flawed graph of 〈|ΔTs+ΔTv|2〉 based on this
approximation together with the graph of 〈|ΔTu+v|2〉, which appears to show that the former is larger than the latter by
a factor ≈2.5 for Δt>50 years. In Fig. we show
the results that we obtain without the approximation. We cannot find any
significant difference between the two graphs (red and blue bullets) for
Δt<300 years; the two curves are entangled, just as in
Fig. d. For Δt>300 years the observed differences are
clearly not statistically significant.
Haar structure functions 〈|ΔTs+ΔTv|2〉 (red bullets) and 〈|ΔTs+v|2〉 (blue bullets).
An alternative, and very simple, estimate for this ratio can be obtained from
the data for the thick red and blue curves in Fig. c, by computing
ΔT's as 50-year moving averages rather than Haar fluctuations. The
standard deviation of ΔTs+ΔTv is
0.072 K, and of ΔTs+v it is 0.060 K, which
yields R≈1.20. This ratio is slightly greater than unity due to the
higher fluctuations in the red graph compared to the blue graph in
Fig. for Δt>300 years. Since this difference on the
longest timescales appears to be a statistical error due to limited sample
size, R=1 is within the error bars of the estimated R (on these timescales
there are only a few independent samples available for estimation of
the variance). If such an error test were crucial, we could have computed the
uncertainty range via a Monte Carlo ensemble of the 1/f noise process, like
we did in Fig. . However, since the two curves are entangled for
Δt>10 years, even very small finite sample size uncertainty will not
allow us to decide that one signal has more power than the other. Moreover,
as will be shown in Sect. , internal variability gives an
additional positive contribution to R which exceeds the error that is
required to explain the estimate R≈1.2 under the linear-response
hypothesis.
The effect of internal variability on the L&V test
The ratio R defined in Eq. () only measures the ratio of
responses if the internal noise is negligible. Hence, even if R were
significantly (in a statistical sense) greater than unity, this increase might
be caused by the internal variability in a model whose response to forcing is
perfectly linear. By using Eq. (), which is valid for a
linear-response model, Eq. () can be written as
R=1+〈|Δε|2〉〈|ΔTs+v|2〉.
This shows that internal noise can increase the rms ratio computed by L&V
even if the response is linear. From the data for the thick brown curve in
Fig. b we have that the standard deviation for the internal noise
Δϵ is 0.03 and hence for Δε a factor
3 larger. The standard deviation of ΔTs+v can be
estimated from the data for the thick blue curve in Fig. c and is
0.06. This yields 〈|Δε|2〉/〈|ΔTs+v|2〉≈0.75, and hence R≈1.32 is the
estimate of the rms ratio based on the linear-response hypothesis.
L&V's arguments against high internal variability
In the first and second drafts of the L&V discussion paper internal
variability was not mentioned. After this problem was raised by us in the
interactive discussion, in the final paper L&V presented two arguments
against the presence of sufficiently high internal fluctuations on the
centennial timescales to explain the raised rms ratio R.
Haar fluctuations for NorESM data. Red curve: Haar fluctuation of
the response to solar+volcanic forcing. Blue curve: the Haar fluctuation
of the summed solar and volcanic response. Magenta curve: Haar fluctuation of
the control run.
The first argument uses the internal variability of the Goddard Institute for Space Studies (GISS) model as an
estimate of the centennial-scale internal variability of the ZC model and
concludes that this estimate is less than 20 % of the total variability
in the ZC model. The authors overlook the fact that the output of the ZC
model is the Niño3 index (temperature anomalies in the tropical Pacific),
while the GISS model output is the average over the northern hemispheric land.
One should also keep in mind that the ZC model was never intended to get the
statistics of variability correct, and so there is no basis for assuming
anything about the magnitude of it relative to GISS. In Fig. 4 of the L&V
paper, fluctuation levels versus scale for ZC and GISS are plotted in the
same panel. For Δt>10 years they almost overlap. However, the ZC
model data are averaged over 100 model runs, so the actual fluctuation level
for the stochastic component is 10 times greater than for the output from
GISS control simulations.
The second argument assumes that the internal noise must have a scaling
exponent β≈0.6, which would yield a negative slope
H=(β-1)/2≈-0.2 of the structure-function plot (see
Fig. d). The actual plot of the structure function of the solar
residual (the yellow circles in Fig. d) has a weakly positive
slope, and hence the authors conclude that the latter is dominated by forced
fluctuations on the centennial to millennium scale. The weakness of this
argument is that it takes as an assumption what the authors want to prove,
namely that internal fluctuations on long timescales are small. It seems
that only long control runs of the ZC model can settle this issue.
Additivity in NorESM data
There are at least three drawbacks with the ZC data. The model is not
representative for the global temperature response, the data analysed has
been averaged over 100 realisations, and L&V had no control runs available
to assess the magnitude of internal variability. They also analysed data from
the NASA GISS E2-R model, but here they lacked the full suite of simulations
with solar-only, volcanic-only, and solar+volcanic forcing, and hence they
could not perform the test of the additivity of responses on a full-blown
GCM. We have acquired a full suite of millennium-long simulations for the
Norwegian Earth System Model (NorESM), which is part of the CMIP5 ensemble. More
specifically, we have analysed solar-only, volcanic-only,
solar+volcanic+anthropogenic, and control runs for the 900-year period 935–1834 CE. We
have omitted the period after 1835 CE to minimise the anthropogenic forcing
in the full forcing simulation, and we treat this as a solar+volcanic
simulation. It is remarkable that all Haar fluctuation curves of all these
signals are almost flat, corresponding to H≈0 or β≈1,
i.e. to a so-called 1/f noise.
In Fig. we have plotted the Haar fluctuations for the
solar+volcanic (total) forcing (red), for the summed responses to solar and
volcanic forcing (blue), and for the control run (magenta). Observe that the
responses to solar and volcanic forcing add up to the response of the
combined forcing. The subadditivity claimed by L&V is completely absent. We
also observe that the internal variability represented by the control run is
quite strong. The standard deviation of the internal variability is two-thirds of
the variability of the signal with solar+volcanic forcing. Moreover, the
internal fluctuations are almost equally strong on long timescales as on
short timescales, contrary to what has been claimed by L&V.
Linearity and intermittencies
The essence of Sect. 4 in the L&V paper is a mathematical result claiming
that linearity in the response implies that the intermittency (the curvature
of the scaling function) is the same for forcing and response. We have a
number of reservations against the application of this result to the data and
the climate models studied in this paper.
The essence of our critique
There are at least three possible sources of different intermittencies of the
forcing and temperatures that are missed in the L&V paper:
The mentioned mathematical result depends on a power-law form of the
linear-response function. On timescales less than a few years, GCM responses
appear to be exponential rather than power law, as shown for the GISS ER-2
model in in Fig. . On the long timescales this assumption is in
direct contradiction to L&V's own claim that GCMs do not reproduce
low-frequency (multicentennial) variability (see also
).
It depends on the perfect power-law scaling of the structure functions
of forcing and response, i.e. that these processes belong to the
multifractal class . This is not true for, e.g., the
volcanic forcing (see Fig. c) nor for GCM responses (see
Fig. ).
The analysis does not account for the internal variability. The
authors have argued that internal variability may be negligible compared to
forced variability on the longest timescales. In Sect. we
demonstrated that this is not the case for GCMs. One should also keep in mind
that for analysis of intermittency, the emphasis is on the smallest timescales.
The intermittency of the temperature signal will be strongly
influenced by, or even dominated by, the internal noise, and hence there is
no reason there should be a strong similarity between intermittencies of
forcing and temperature in a linear-response model.
Grey curve is the global temperature response to a sudden
quadrupling of atmospheric CO2 concentration in the GISS E2-R model. Blue curve
is a fit of superposition of two exponential responses (two-box model
solutions), with the two exponential time constants being τ1=1.3 years and
τ2=176 years. Red curve is a power-law fit and is a poor fit up to
several years.
(a) A zoom-in on the volcanic forcing signal shown in
Fig. a. (b) The ACF estimated for the volcanic forcing
signal. (c) The structure functions (empirical moments)
S^q(Δt) for the volcanic forcing signal estimated for
q=(0.2,0.4,…,4.0). The red dashed line is a linear fit to the
log–log plot of S^2(Δt). (d) The scaling function
ζ(q) computed from linear fits to the S^q(Δt)'s over the
interval Δt∈(4,128). The observation that ζ(2)≈1
suggests that the process is uncorrelated on these timescales.
Effect of imperfect power laws on intermittencies
Here we present some theoretical considerations which demonstrate that imperfect
scaling (power laws) of the response kernel and the structure functions can lead
to different intermittency of forcing and response in a linear-response model. In
Sect. we demonstrate this by an example, so the present
subsection can be skipped by readers who are only interested in such a demonstration.
The general linear response model Eq. () can be written as a convolution
of the forcing F(t) with a response kernel G(t):
L^[F(t)]=∫-∞∞G(t-t′)F(t′)dt′.
For a general analysis of moments it is convenient to formulate the moments
in the frequency domain rather than the time domain. Thus, we
Fourier-transform Eq. () to write
T(f)=G(f)F(f),
where T(f), F(f), and G(f) are the Fourier
transforms of T(t), F(t), and G(t), respectively. By defining structure
functions in frequency domain, SqT(f)≡〈|T(f)|q〉, SqF(f)≡〈|F(f)|q〉,
we have the general linear-response model formulated as a linear relation
between forcing and response structure functions of order q in the
frequency domain, with the ensemble average of the qth power of the
transfer function |G(f)| as a constant of
proportionality:
SqT(f)=〈|G(f)|q〉SqF(f).
L&V assume a power-law linear response. This corresponds to a response
function of the form
G(t)=ξ(t/μ)H-1/2θ(t),
where ξ=1km2J-1, μ is a constant in units of time
which characterises the strength of the response, H is the scaling exponent
for the response used by L&V, and θ(t) is the unit step function. The
Fourier transform of this response function yields (see )
|G(f)|=ff0-(H+1/2),
where
f0=ξμΓ(H+1/2)1H+1/22πμ,
and Γ(x) is the Euler gamma function. Hence the L&V special case of
Eq. () is
SqT(f)=ff0-q(H+1/2)SqF(f).
The next assumption made by L&V is that both forcing and response exhibit
multifractal scaling. If we write the structure functions as (dropping the
superscripts)
Sq(f)=Cq(f)f-η(q),
the multifractal scaling assumption is that the multiplicative factor
Cq(f) is independent of the frequency f, such that the structure
functions are perfect power laws in f. This is a very
restrictive assumption that is not satisfied by any of the data in this
study. If Eq. () holds true, a plot of logSq(f) vs.
logf is linear with slope -η(q). The essence of the L&V approach
(although some technicalities differ) corresponds to fitting the logSq(f) vs. logf curves with straight lines at the highest
frequencies, or, in other words, drawing tangent lines to the curves at the
Nyquist frequency fN. The negative slopes of these lines are interpreted
as the scaling functions η(q). This corresponds to defining the scaling
functions by
η(q)=dSq(f)d(logf)f=fN,
and from Eq. () we then find the fdependence of Cq(f) which
represents the deviation from multifractal scaling. The L&V approach
includes normalizing the signals T(t) and F(t) such that they have the
same power at the lowest frequency f=1, i.e. S2T(1)=S2F(1). If
H≠-1/2, then Eq. () implies that f0=1; putting f=1 in
Eqs. () and () we find
SqT(1)=SqF(1)=CqT(1)=CqF(1)
for all q. From the logarithm of Eqs. () and () we
find, for f> 1,
ηF(q)-ηT(q)+q(H+1/2)=log[CqF(f)/CqT(f)]logf.
If T(t) and F(t) exhibit perfect multifractal scaling, we have
Cq(f)=Cq(1), and from Eq. () the right-hand side of
Eq. () vanishes. Hence, for this case we have the L&V results that
the curves ηT(q) and ηF(q) have the same curvature; i.e. the
response and forcing exhibit the same multifractal intermittency. However,
the term q(H+1/2) on the left-hand side arises from the particular
power-law form of the linear-response function shown in Eq. ().
With another form of the linear-response kernel this term might not be linear
in q, and this could introduce different curvature of ηT(q) and
ηF(q). Different curvature is also introduced if the structure
functions are not perfect power laws. Then the term on the right of
Eq. () will in general not vanish, and it may have a non-zero
second derivative. This may give rise to different curvatures of ηT(q)
and ηF(q) even if the response is linear with the power-law response
kernel given by Eq. ().
Response to volcanic forcing
An important point in L&V is that intermittency in volcanic forcing and the
corresponding temperature response are different, and that this is a
signature of nonlinearity in the response. In this subsection we shall first
demonstrate that the intermittency in the volcanic forcing is not
multifractal; i.e. all the structure functions are not power laws. This is a
symptom of the lack of correlations between bursts that characterises a
multiplicative cascade. Next, we shall show by using L&V's trace-moment
analysis on a simple linear-response model that we can reproduce the
intermittency observed in the response to volcanic forcing in the ZC model.
This linear response exhibits a similar power spectrum, similar trace
moments, and almost identical intermittency parameters to the ZC response.
And more importantly, these features are considerably different in the forcing
and the response, even though the response model is linear. It demonstrates
that these results obtained from the ZC model are not a signature of
nonlinearity in the response.
Let us first build some intuition on the nature of the volcanic forcing. In
Fig. a we have zoomed in on the volcanic forcing signal used in the
ZC model. Each volcanic eruption is represented by two–three data points (years)
different from zero (some large eruptions are represented by a few more
points). If the eruptions are distributed randomly in time (Poisson
distributed) the autocorrelation function (ACF) will vanish after a time lag
of a few years. This is exactly what we observe in Fig. b. The
spectral structure functions used in Sect. are convenient
for theoretical studies but not for estimation based on short and spiky time
series. Here it is better to use the standard structure functions which are
computed from the empirical moments:
S^q(Δt)=(N-Δt)-1∑t=1N-Δt|Y(t+Δt)-Y(t)|q,
where Y(t)=∑t′=0tF(t′) is the cumulative sum of the forcing time
series. This is a standard estimator commonly used in analysis of stationary
time series. It is much more transparent than the trace moments employed by
L&V and contains no hidden assumptions about power-law structure functions
or the existence of an “outer scale” for these power laws (see discussion
in Sect. ). The empirical
moments of the volcanic forcing signal are shown in Fig. c. The
steeper slopes (slope ≈q) for Δt≤4 are due to the
smoothness of the forcing signal on these short timescales, signified by the
ACF in Fig. b. For q=2 the structure function looks quite
straight and with slope close to 1 in the log–log plot for the scale range
4–100 years. For smaller q the plots become more curved. This is
symptomatic for a stationary, uncorrelated process (Lévy process) which
is non-Gaussian on short timescales, although the central limit theorem
requires that it converge to a Gaussian on the longer scales. According to
, such a process is not multifractal (see also Sect. 2.5 and
Appendixes A–C in ). In practice,
L&V's approach corresponds to assuming that the moments can be written in
the power-law form S^q(Δt)∼Δtζ(q), where the
scaling function ζ(q) is estimated by fitting straight lines to the
structure functions in the log–log plot in the range 4–100 years. This has
been done in Fig. d. The curved scaling function is interpreted by
L&V as a signature of multifractality, but this interpretation is correct
only if all structure functions are power laws (straight lines in log–log
plots). It is easily demonstrated that very similar results are obtained by
random shuffling of the onset times of the volcanic spikes, which would
convert a multifractal signal into a realisation of a Lévy process. If
the original signal were a multifractal, the result should be quite different
after shuffling. For a deeper discussion of these disagreements see the
interactive discussion and in particular our author comment AC3
.
Our main focus here, however, is not on the incorrect multifractal
interpretation of the scaling analysis, but on the incorrect conclusions
drawn from this analysis when it comes to nonlinearity in the response. As a
means to investigate this point we construct a linear-response model that
mimics the ZC response to the volcanic forcing. The ZC response is shown by
the blue curve in Fig. a. We observe that every volcanic spike
seems to be succeeded by a damped oscillation. Thus, we construct a linear,
damped harmonic oscillator response model and select the parameters to
produce a response signal that looks similar to that of the ZC response to
the volcanic forcing when we drive the model with stochastic forcing in
addition to the volcanic forcing. We make no attempts to fine-tune the model
parameters, since this extremely simple model obviously is not an accurate
substitute for the ZC model. The purpose of devising this model is only to
demonstrate that a linear model can produce a response with intermittency
parameters very different from those of the forcing. These are results which
L&V contend can only arise from nonlinearity of the response.
(a) Blue curve is the average over 100 realisations of the
response to volcanic forcing in the ZC model, and the red curve is the
response to this forcing plus a stochastic Gaussian white-noise forcing in a
linear, damped harmonic oscillator model. (b) Result of trace-moment
analysis of the volcanic forcing signal. It is very similar to the
corresponding panel in Fig. 6 of L&V. (c) Result of trace-moment
analysis of the harmonic oscillator response shown by the red curve in
panel (a). It is very similar to the corresponding panel for the ZC
response to volcanic forcing in Fig. 6 of L&V.
Analysis of global temperature responses in the NorESM model.
(a) The volcanic forcing (red) normalised such that the largest
spikes are approximately equal to the spikes of the response signal (black).
(b) The red curve in (a)+ the control-run temperature
signal. (c) Structure functions (of cumulative sum) of volcano
forcing. (d) Scaling function derived from (c).
(e) Structure functions of the response to volcanic forcing.
(f) Scaling functions derived from (e).
(g) Structure functions of the signal in (b).
(h) Scaling function derived from (g). The red line arises
from fitting straight lines in the entire scale range plotted, 4–128 years.
The blue line is from fitting only in the scale range 16–128 years. It shows
weak intermittency in both cases, as well as that estimated intermittency
depends on the scale range chosen for fitting. The difference in curvature
(reduction of intermittency) between (d) and (h) is
exclusively caused by the addition of the internal noise represented by the
control run, and the similarity between (f) and (h)
indicates that the internal variability is the main cause of reduced
intermittency in the response to volcanic forcing.
The response according to the linear model is shown by the red curve in
Fig. a, and we compute the trace moments and intermittency
coefficients for this linear-response signal. We have used the Mathematica
routines downloaded from Shaun Lovejoy's web page for these computations to
make sure that the results are comparable to those presented by L&V.
Fig. b is a reproduction of Figure 6a, top right, in L&V for the
volcanic forcing. L&V interpret the wide spread in slopes of the
trace-moment curves as signature of multifractal intermittency, and they
compute the intermittency coefficients C1=0.48 and α=0.31 (their
Table 1). The results depend on the exact fitting range chosen, so we cannot
expect to get exactly the same results for these parameters. We find
C1=0.52 and α=0.13 (which makes us wonder if α=0.31 in L&V
is a misprint). In Fig. c we have computed the trace moments for an
arbitrary realisation of the linear-response model. This figure is very
similar to their Figure 6a, bottom right, for the ZC response. The
intermittency parameters computed by L&V for this case are C1=0.054 and
α=2.0, while our results for the linear model are C1=0.039±0.013
and α=1.92±0.03. These numbers are mean values over an ensemble of
100 realisations of the linear-response model, and the errors are ±2σ, where σ is the standard deviation over this ensemble. The
important feature here is not the similarity between the intermittency
parameters for the ZC model and this linear model but rather the great
difference in these parameters between volcanic forcing and response in the
linear model. L&V interpret this difference as a signature of nonlinearity,
but our exercise shows that such a difference can be obtained from a simple
linear-response model with internal noise.
Intermittency in GCMs
The breakdown of condition (III) due to internal variability in GCMs is
clearly illustrated in Fig. , which is based on the data from the
NorESM model. Figure a shows the volcanic forcing signal (red) and
the model response to this signal (black). Figure b shows a signal
composed of two components; one is the volcanic forcing signal normalised
such that the magnitudes of the large volcanic spikes roughly match those of
the volcanic response signal. This signal can be thought of as the
instantaneous response to the stochastic forcing. The other component is the
internal variability represented by a control run. This composite signal
represents a trivial linear transformation (multiplication by a normalisation
factor) plus a signal representative for the internal variability.
Figure c and d show the SFs and the scaling
function for the volcanic forcing computed from straight lines fitted to the
SFs in the range displayed in Fig. c. Figure e and f show
the same for the model response signal to the volcanic forcing, and
Fig. g and h for the composite signal shown in Fig. b.
According to L&V (who believe condition III is irrelevant), the
intermittency shown by the curvature of the scaling function in
Fig. d should be preserved in the scaling function for the
composite signal shown in Fig. h, but it is not. The latter signal
is almost non-intermittent due to the “contamination” from the internal
noise. The contamination explains the reduced intermittency observed in the
response to the volcano forcing shown in Fig. e and f. This proves
that nonlinearity in the response is not required to explain the difference
in intermittency between forcing and response in GCMs.
Discussion and conclusions
L&V conclude
from their analysis of additivity that nonlinearity in the form of
subadditivity is strong primarily on timescales longer than 50 years and
that there are specific physical reasons for this, like temperature albedo
feedbacks. Our comment is that we find no reason why responses should be more
linear on short than on long timescales, in particular not the response to
the burst-like volcanic forcing. The response of local climatic variables on
synoptic and seasonal scales to strong volcanic eruptions is certainly
nonlinear. But on longer timescales, the global temperature will change in
proportion to the change in heat content in the upper ocean, which again will
change in proportion to the net radiative flux. The response in the presence of
feedbacks that modify the radiative flux is not generally expected to become
nonlinear. Feedbacks are typically modelled linearly, although in some cases
different feedbacks may combine nonlinearly.
The ENSO phenomenon is probably a nonlinear mode in the climate system and
is part of the internal variability, even though it can be influenced by
external forcing. The nonlinear nature of the oscillation makes it likely
that the timing of El Niño events can be influenced by external forcing
such as strong volcanic eruptions. In general, the modes of internal
variability of the climate system are results of nonlinear processes, and the
modes are probably responding nonlinearly to external forcing. But we find it
less likely that the ensemble-averaged global temperature response is
nonlinear to an extent that is detectable.
On the other hand, the intermittency analysis by L&V is designed to detect
nonlinearity on short timescales, so it appears that the nonlinearity they
claim to detect by this analysis is different from the subadditivity on long
timescales. The trace-moment analysis employed is rooted in ideas of
intermittency and multifractality, which have emerged from turbulence theory.
It was used by in the context of rain and cloud fields, but
more recently they have worked extensively to extend the ideas of turbulent,
multiplicative cascades, not only to atmospheric dynamics and weather but
also to climate dynamics across a vast range of scales
. The validity of extending the
turbulence framework to encompass the dynamics of the entire climate system
across the scales is not obvious and deserves to be challenged. The simple
linear energy-balance modelling is one example of an alternative framework.
Models of this kind can be extended to incorporate several interacting
subsystems with different response times (multi-box models) and can give
rise to responses that are close to power laws over a certain range of scales
(see Fig. ). But there are also other competing paradigms based on
treating the climate as a high-dimensional dynamical system residing in
non-equilibrium stationary states, and invoking response theory of
non-equilibrium statistical mechanics for prediction of changes in the
globally averaged surface temperature as well as its spatial patterns (see
, and references therein).
Tests formulated on the basis of one particular theoretical framework run
the risk of becoming self-fulfilling. The trace-moment analysis employed by
L&V explicitly assumes the existence of multifractal scaling up to a certain
outer scale, and lines are fitted to the trace moments under the
constraint that they all cross at this outer scale. The slopes of these lines
are used to compute the intermittency parameters, even in cases where these
lines are poor fits to the actual trace moments. The method is automatised
and contains no means to discriminate between true multifractal and
non-Gaussian uncorrelated processes (Lévy processes). The implication of
failing to make this distinction is that a mathematical result for
multifractal processes (stating that a linear transformation preserves
intermittency) is applied by L&V to processes for which this result is not
valid.
The main conclusions of this comment are the following: a correct treatment,
without unjustified approximations, of the issue of additivity in the
Zebiak–Cane model gives no reason for rejection of a linear-response model
(see Fig. ). This conclusion holds even without accounting for
internal variability but is enforced by the inclusion of this effect. This
was demonstrated in Sect. by the alternative test
introduced in Sect. .
L&V's analysis of intermittencies is based on a mathematical result which
states that if the response is linear the intermittency computed through
trace-moment analysis must be the same in forcing and response. However, this
result holds only if both forcing and response belong to the class of
multifractals, i.e. if all structure functions are power laws
, and in addition it requires that the response function is
a power law in the entire scale range of interest. Figure
demonstrates the structure functions of volcano forcing are not power laws,
and Fig. that this is the case also for structure functions of
global temperature in GCMs. Figure shows that the temperature
response function in GCMs is not a power law on all available timescales.
In Fig. we illustrated by an example that the intermittencies can
be very different in forcing and response produced by a linear-response model
with internal variability. The role of internal variability in reducing
the intermittency in the linear response to an intermittent forcing was
demonstrated for GCM data in Sect. and displayed
in Fig. . Hence, our conclusion is that the intermittency analysis
of L&V does not constitute a valid test for rejecting the linear-response
hypothesis.
Acknowledgements
This paper was supported by the Norwegian Research Council (KLIMAFORSK
programme) under grant no. 229754. We are grateful to Shaun Lovejoy for
providing Mathematica codes for estimation of Haar structure functions and
trace moments on his web page
(http://www.physics.mcgill.ca/~gang/software/index.html) and to
Hege-Beate Fredriksen for debugging these codes. Odd-Helge Otterå has kindly
made available to us the data from the NorESM model. We also thank two reviewers
for constructive comments and useful suggestions for improvement of the
manuscript.
Edited by: B. Kravitz
Reviewed by: three anonymous referees
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