# Does the problem of global warming exist at all?
Insight from the temperature drift induced by inevitable colored noise

V.D. Rusov1111Corresponding author: Vitaliy D. Rusov, E-mail<EMAIL_ADDRESS>V.P. Smolyar1, M.V. Eingorn1,2,
T.N. Zelentsova1, E.P. Linnik1, M.E. Beglaryan1, B. Vachev3

###### Abstract

In the present paper we state a problem of the colored noise nonremovability
on the climatic 30-year time scale, which essentially changes the angle of
view on the known problem of global warming.

1Department of Theoretical and Experimental Nuclear Physics,
Odessa National Polytechnic University, 1 Shevchenko ave., Odessa 65044,
Ukraine

2CREST and NASA Research Centers, North Carolina Central University,
Fayetteville st. 1801, Durham, North Carolina 27707, U.S.A.

3Institute for Nuclear Research and Nuclear Energy, Tsarigradsko Chaussee
Blvd. 72, Sofia 1784, Bulgaria

##  1 Introduction

It is known that the historical surface temperature data set HadCRUT provides
a record of surface temperature trends and variability since 1850. The most
well established version of this data set, HadCRUT3 [2], has been produced
benefiting from recent improvements to the sea surface temperature data set
which forms its marine component and from improvements to the station records
which provide the land data. A new version of this data set, HadCRUT4 [3],
improves and updates the gridded land-based Climatic Research Unit temperature
database and virtually does not differ from its predecessor – HadCRUT3 [2].

In the framework of both versions, basing on a careful analysis, a
comprehensive set of uncertainty estimates has been derived to accompany the
data and the following perfectly clear conclusion has been made: ”…Since the
mid twentieth century the uncertainties in global and hemispheric mean
temperatures are small, and the temperature increase greatly exceeds its
uncertainty. In earlier periods the uncertainties are larger, _but the
temperature increase over the twentieth century is still significantly larger
than its uncertainty_” (Fig. 1a).

Figure 1: HadCRUT3 global temperature time series at annual resolution (blue)
and smoothed (orange) annual resolution, obtained via averaging of the blue
data with (a) 21-year and (b) 30-year moving intervals.

At the same time, analyzing the methodology of the temperature data
reconstruction given in these rigorous researches [2, 3], it is necessary to
note one important, in our opinion, key point about these papers. It consists
in the complete absence of the commentaries on the physical reasons of
ignoring the nature and type of the noise accompanying the procedure of
successive temperature averaging on different time scales. This is a major
point, since every time scale the temperature of the global climatic system is
being averaged on, is characterized, in general case, by its own type of the
observed noise, which is known to be determined by the spectral density of the
temperature fluctuations in the studied record. In other words, switching from
one time scale (monthly) to another (annual or thirty-year) requires the
knowledge of the corresponding spectral density of the temperature
fluctuations in order to estimate the most important quantity – the
temperature variance. Meanwhile, the averaged temperature variance on the
given time scale may be characterized _in certain cases_ by a _nonremovable_
variance of the colored noise.

The effect of colored noise variance nonremovability on a given time scale
means just that the colored noise is generated by the temperature fluctuations
in the climatic system. Therefore, its variance is actually a temperature
variance equivalent on this time scale. Alternatively speaking, a nonremovable
noise is not an ”interference” masking the real temperature, but rather a
characteristic measure of the temperature fluctuations on a given time scale,
which cannot be disregarded or ”cleaned out” by some kind of time-averaging
procedure.

The analysis of conditions and consequences of such effect of the colored
noise nonremovability typical to the power spectrum of the earth climatic
system (ECS) temperature fluctuations around tricennial time scale (the
climatic state 222Climate is commonly defined as a statistical ensemble of ECS
states characterized by a corresponding set of thermodynamical parameters
(temperature, pressure, etc.) averaged over a long period of time. The
classical period is 30 years, as defined by the World Meteorological
Organization. Hence, it follows that the weather deviation from the climatic
norm cannot be considered as a climate change.) is the primary goal of this
Letter.

##  2 Allan variance and a temperature drift induced by colored noise

In order to retrieve the ”true” variance magnitude for the colored noise of
$1/f^{\alpha}$ type (where $\alpha\geqslant 1$) which dominates in ECS in the
climate state, it is necessary to find a good equivalent of the ”original”
virtual temperature record in some way. Such equivalent record must
characterize the mean global trend while not being ”cleaned” by any special
time-averaging procedures such as the ones used for HadCRUT3 database [2, 3].

For this purpose we calculated the Allan variance [1, 4, 6, 7] for every
historical temperature record observed at 5113 meteorological stations over
the world as used in the HadCRUT3 calculations

$\sigma_{A}^{2}(\tau)=\frac{1}{2}\left\langle\left(y_{i+1}(\tau)-y_{i}(\tau)\right)^{2}\right\rangle,$
(1)

which is a variance of the first differences averaged over the time interval
$\tau$ of the signal $y$. The Allan variance behavior depends on the form of
the noise power spectral density (PSD). For a noise with PSD of the
$1/f^{\alpha}$ form the Allan variance (1) is proportional to
$\tau^{\alpha-1}$, where $\alpha=0$ stands for white (uncorrelated) noise,
$\alpha=1$ for ”$1/f$” (flicker) noise, and $\alpha\geqslant 2$ for correlated
low frequency (drift) noise [1, 4, 6, 7].

Figure 2: Allan variance plot for a set of temperature time series, obtained
at the 5113 meteorological stations over the world. The green line corresponds
to a temperature record at Geneva-Cointrin (Switzerland) weather station. The
color lines reproduce the isoclines, i.e. the lines of equal density of points
that form the general ”relief” of the processed weather stations data.

Analysis of the Fig. 2 demonstrates that the experimental temperature record
obtained at Geneva-Cointrin weather station over the years 1753 – 2011 is a
sufficient equivalent of the ”original” virtual temperature record for our
purposes, since it matches the mean global trend well and is not ”cleaned” by
any kind of time-averaging procedures used in HadCRUT3 [2, 3].

Basing on the comparative analysis of Geneva-Cointrin and HadCRUT3 global
temperature time series, performed by means of Allan variance (Fig. 3) and PSD
(Fig. 4), one may conclude the following. It is obvious, for example, that the
time-averaging procedures used in HadCRUT3 analysis suppressed the Allan
variance and PSD of the temperature fluctuations completely (relative to the
original Geneva-Cointrin data). In addition to that they changed the noise
type, i.e. the white noise to the flicker noise (see the yellow area on Fig. 3
and Fig. 4a), while the original flicker ($1/f$) and drift ($1/f^{2}$) noises
survived, but were highly depressed (see the green and ping areas on Fig. 3
and Fig. 4b). On the other hand, the original drift noise variance apparently
reaches the value of $\sigma_{A}^{2}$ (see the star at Fig. 3) around the
tricennial time scale (the climatic state).

Figure 3: Allan variance plot for temperature records, obtained at Geneva-
Cointrin weather station (the green line) and within the framework of HadCRUT3
(the blue line). Figure 4: PSD of temperature fluctuations for (a) HadCRUT3
and (b) Geneva-Cointrin global temperature time series. The insets are laid
over the PSD of temperature fluctuations adopted from [5].

The latter means that if the drift noise around the tricennial time scale (see
the star at Fig. 3) is nonremovable when ECS is in a state of _climate_ , the
standard deviation $\sigma_{A}$ reaches the value of $\sim 0.45$ and
definitely should be taken into account when constructing the HadCRUT3 global
temperature time series at smoothed annual resolution (the yellow line on Fig.
1a). Its adoption obviously may change the interpretation of HadCRUT3 analysis
drastically (see Fig. 1b, the gray color).

Considering the extreme importance of such conclusion, let us discuss the
possible physical reasons of the colored noise nonremovability in climatic
states description below. For this purpose let us examine a point system,
e.g., ”water – vapor”, with two phase transitions taking place in it, having
the interacting order parameters $X$ and $Y$. At a point of phase transition
lines intersection a potential of such system may be written down in the form
of expansion [11, 13, 12]:

$\Phi=\Phi_{0}-\alpha_{1}X^{2}-\alpha_{2}Y^{2}-\alpha_{12}XY+\beta_{1}X^{4}+\beta_{2}Y^{4}+\beta_{12}X^{2}Y^{2}.$
(2)

Introducing the fluctuating forces in the form of additive terms
$\Gamma_{1}(t)$ and $\Gamma_{2}(t)$, where $\Gamma_{1}(t)$ and $\Gamma_{2}(t)$
are the Gaussian delta-correlated noises, it is possible to pass to a system
of coupled Langevin equations, following [11, 13, 12]:

$\partial X/\partial
t=-2\beta_{12}XY^{2}-4\beta_{1}X^{3}+2\alpha_{1}X+\alpha_{12}Y+\Gamma_{1}(t),$
(3)

$\partial Y/\partial
t=-2\beta_{12}YX^{2}-4\beta_{2}Y^{3}+2\alpha_{2}Y+\alpha_{12}X+\Gamma_{2}(t).$
(4)

A plan for solution of such system is rather simple. The system (3)-(4) is
solved numerically using, for example, the Euler method with different
parameters. The obtained numerical solutions $X(t)$ and $Y(t)$ then undergo a
Fast Fourier Transform and result in a spectral density of fluctuations. In a
simplest case, when the solutions have divergent spectral characteristics, the
parameters of the system, according to [11, 13, 12], are $\beta_{12}=1/2$,
$\beta_{1}=\alpha_{12}=1$, $\alpha_{1}=\alpha_{2}=\beta_{2}=0$, and the system
(3)-(4) takes on the following form:

$\partial X/\partial t=-XY^{2}-4X^{3}+Y+\Gamma_{1}(t),$ (5)

$\partial Y/\partial t=-YX^{2}+X+\Gamma_{2}(t).$ (6)

In the absence of the external noise the solution has asymptotics $X(t)\to
t^{-1/2}$ and $Y(t)\to t^{1/2}$, when $t\to\infty$. The results of numerical
modeling presented in [11, 13, 12] show that the frequency dependence of the
spectral density $S_{X}(f)$ of fluctuations $X(t)$ has the form of
$1/f^{\alpha}$, where $\alpha\approx 1$. In other words, the system (3)-(4)
generates the $1/f$-noise, i.e. the flicker noise. Meanwhile, the spectral
density of the order parameter $Y(t)$ fluctuations depends on the frequency
like $S_{Y}(f)\propto 1/f^{\mu}$, where $1.5\leqslant\mu\leqslant 2$, i.e.
generates the drift noise.

The potential in which the system performs a random walk described by the
system (5)-(6) has the form

$\Phi=\Phi_{0}-XY+X^{4}+(1/2)X^{2}Y^{2}.$ (7)

As follows from (7), this is a double-well potential or, more precisely
speaking, a two-valley potential, i.e. a potential surface has two valleys. In
the absence of the external noise the phase trajectory of a system, depending
on the initial conditions, is enclosed entirely in one of the valleys. In the
presence of an external small-amplitude noise the system performs a random
walk inside the valley. Increasing the noise intensity leads to the system
jumps from one valley to the other, and $X(t)$ displays the $1/f$-noise, while
the order parameter $Y(t)$ displays a $1/f^{2}$-noise.

Thereby this example indicates that the origin of the heat pulsations with a
spectral density inversely proportional to a frequency may be related to the
interaction between the nonequilibrium phase transitions in the system of two
coupled Langevin equations which transforms the white noise into two
oscillation modes with spectral densities proportional to $1/f$ and $1/f^{2}$
[11, 13, 12].

The intersection and interaction of two phase transitions is a rather
widespread phenomenon. Therefore, the model (3)-(4) [6] may be universal
enough to serve as a basis for the explanation of the nonremovable colored
noise formation mechanism in a wide range of processes involving the
nonequilibrium phase transitions. Let us show this for a point nonlinear
system ”water – vapor” in the particular climatic model below.

##  3 Climatic two-well potential and bifurcation model of the Earth global
climate

In our earlier papers we proposed a bifurcation model for the energy balance
of the Earth global climate. Its theoretical solution was in a good agreement
with the known experimental data on Earth surface paleotemperature evolution
for the last 420 thousand years and 740 thousand years, obtained within the
antarctic projects Vostok and EPICA Dome C respectively. In the framework of
the proposed model a concept of climatic sensitivity was also introduced,
which manifests a property of temperature instability in the form of the so-
called hysteresis loop, as shown in [9, 10]. Basing on this concept and using
the bifurcation equation of the model, we reconstructed a time series for the
global ice volume for the last 1 million years which fits well the
experimental time series of $\delta^{18}O$ concentrations in marine sediments.
A base for this result lies in a fact that an effective mechanism of climatic
”cold-warm” oscillations formation had been built into the model and was
provided by the interaction of the nonequilibrium phase transitions in a
”fresh water – water vapor” system in the Earth boundary layer [9, 10].

If we add to the stated above that our bifurcation model of the Earth global
climate, developed in [11, 13, 12], was built upon a climatic double-well
potential

$U(T,t)=\frac{1}{4}T^{4}+\frac{1}{2}a(t)T^{2}+b(t)T,$ (8)

where

$a(t)=-\frac{1}{4\delta\sigma}a_{\mu}H_{\oplus}(t),$ (9)

$b(t)=-\frac{1}{4\delta\sigma}\left[\frac{\eta_{\alpha}S_{0}}{4}+\frac{1}{2}\beta+\frac{1}{2}b_{\mu}H_{\oplus}(t)\right],$
(10)

which is qualitatively identical (given $Y(t)\sim\mathrm{const}$) to a two-
valley potential (7), it becomes clear that ECS is really capable of
generating the colored noise which is not actually a masking ”interference”,
but rather an internal property of the ECS temperature pulsations. Here
$H_{\oplus}$ is the Y-component 333The physical sense of the Y-component of
the Earth magnetic field adoption is discussed in [8] in detail. of the Earth
magnetic field intensity; $T$ is the mean global temperature of the Earth
surface at the time $t$; $S_{0}=1366.2~{}W/m^{2}$ is the solar constant;
$\delta=0.95$ is the emissivity of the Earth surface; $\sigma=5.67\cdot
10^{-8}~{}W/(m^{2}K^{4})$ is the Stefan-Boltzmann constant;
$\eta_{\alpha}=0.01513~{}K^{-1}$, $a_{\mu}=0.5398~{}W/(m^{2}K^{2})$,
$b_{\mu}=-310~{}W/(m^{2}K)$; finally, $\beta=0.006~{}W/(m^{2}K)$ is the carbon
dioxide accumulation rate in the atmosphere normalized by a unit temperature.

Following this conclusion, let us consider a basic equation of the bifurcation
model of the Earth global climate for the average Earth surface temperature on
the 30-year time scale [8, 9, 10]:

$\frac{m^{*}}{4\delta\sigma}\frac{dT}{dt}=T^{3}+a(t)T+b(t)+\xi(t),$ (11)

where the numerical value $0.129$ was used for $m^{*}$, and

$\langle\xi(t)\rangle=0,~{}~{}~{}\langle\xi(t)\xi(t^{\prime})\rangle=2D\delta(t-t^{\prime}).$
(12)

Obviously, the solution of the stochastic differential equation (11) fits into
the total uncertainty limits of HadCRUT3 global temperature time series at
smoothed annual resolution completely (Fig. 1b, the red line), where a
nonremovable noise variance predominates. At the same time, as the numerical
integration of Eq. (1) using the Runge-Kutta fourth order method with
different initial conditions reveals, the obtained solution (Fig. 1b, the red
line) is strongly stable. It means that in the presence of a nonremovable
colored noise on the climatic 30-year time scale the interpretation of the
HadCRUT3 analysis data changes drastically, replacing the known ”global
warming” paradigm with an alternative theory of climate as a highly stable ECS
state which is characterized by a temperature $\sim 287~{}K$ within the
colored noise variance.

##  4 Conclusion

One of the major results of the present paper is a statement of the problem of
the possible nonremovability of the colored noise on the climatic 30-year time
scale, which changes the angle of view on the known problem of global warming
radically.

One of the basic climate-generating nonlinear ”fresh water – water vapor”
subsystem was involved to explain the mechanism of the nonremovable colored
noise formation. The appearance of the nonremovable thermal pulsations with
the spectral density inversely proportional to the frequency is explained by
the interaction of the nonequilibrium phase transitions in such system. In
other words, is has been shown that ECS is really capable of generating the
nonremovable colored noise which is an internal property of the temperature
pulsations and not just a masking ”interference”.

However it must be admitted that all the arguments adduced here apply not to a
direct, but to an indirect proof of the colored noise nonremovability on the
climatic 30-year time scale. This is also true for our model representations,
which, although have passed a reliable verification by fitting the known
experimental paleotemperature data on a large time scale, have the same
arguments of indirect action. At the same time it should be noted that the
problem of the possible noise nonremovability formulated in the present paper
makes any claims about global warming inappropriate until the problem of the
colored noise origin on the climatic 30-year time scale is definitely solved.

## Acknowledgements

This work is partially supported by EU FP7 Marie Curie Actions, SP3-People,
IRSES project BlackSeaHazNet (PIRSES-GA-2009-246874).

The work of M. Eingorn was supported by NSF CREST award HRD-0833184 and NASA
grant NNX09AV07A.

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