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 paper by

According to a widely accepted principle in the philosophy of science

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

After these remarks we are ready to formulate the

For realistic strength of the global forcing the statistics of the
internal variability

The global temperature can be expressed as a sum of this internal
variability and a linear response to the forcing, i.e.

The data used from the ZC model are the temperature (more precisely, the
Niño3 index) after averaging over 100 simulations with the same forcing

The next step is to produce a fluctuation

Figure

If the linear-response hypothesis is true, we can determine

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.

We have now distinguished the internal noise from the solar-driven
temperature signal by means of the very simple linear-response assumption,
Eq. (

The orange bullets in Fig.

If the characterisation we have made of the internal noise is correct, and
the linear hypothesis is true, then Eq. (

In Fig.

Brown bullets: Haar fluctuation function of

This test would have been stronger if we had had a more direct estimate of the
internal variability. In an interactive comment (SC3),

The L&V test of additivity shown in their paper is simpler than described
in Sect.

Haar structure functions

An alternative, and very simple, estimate for this ratio can be obtained from
the data for the thick red and blue curves in Fig.

The ratio

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

Haar fluctuations for NorESM data. Red curve: Haar fluctuation of
the response to solar

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

The second argument assumes that the internal noise must have a scaling
exponent

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

In Fig.

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.

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.

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

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.

Grey curve is the global temperature response to a sudden
quadrupling of atmospheric

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.

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.

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.

Analysis of global temperature responses in the NorESM model.

The response according to the linear model is shown by the red curve in
Fig.

The breakdown of condition (III) due to internal variability in GCMs is
clearly illustrated in Fig.

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

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.

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

In Fig.

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
(