GALACTIC COSMIC RAYS – CLOUDS EFFECT AND BIFURCATION MODEL
OF THE EARTH GLOBAL CLIMATE.
PART 2. COMPARISON OF THEORY WITH EXPERIMENT.
V. Rusov1,2,3, A. Glushkov4, V. Vaschenko 3 , О. Mihalys1, S. Kosenko1, S. Mavrodiev 5, B. Vachev5
1
National Polytechnic University, Odessa, Ukraine
2
Bielefeld University, Bielefeld, Germany
3
National Antarctic Center, Kiev, Ukraine
4
5
Odessa State Environmental University, Odessa, Ukraine
Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria
Abstract
The solution of the energy-balance model of the Earth’s global climate proposed in Ref. [1] is
compared with well-known experimental data on the palaeotemperature evolution of Earth’s surface over
past 420 kyr and 740 kyr obtained in the framework of Antarctic projects the EPICA Dome C and
Vostok.
The Solar-Earth mechanism of anomalous temperature jumps observed in the EPICA Dome C and
Vostok experiments and its relation with the “order-chaos” transitions in convection evolution in the
liquid Earth core responsible to the mechanism of the Earth magnetic field inversions was discussed. The
stabilizing role of the slow nuclear burning on the boundary of the liquid and solid phases of the Earth's
core (georeactor with power of 30 TW) for convection evolution in the liquid Earth’s core and hence in
the Earth’s magnetic field evolution is pointed out.
In the framework of proposed bifurcation model (i) the possibility of sharp warmings of the
Dansgaard-Oeschger type during last glacial period due to stochastic resonance is theoretically argued,
(ii) the concept of the climatic sensitivity of water (vapour and liquid) in the atmosphere, which reveals
the property of temperature instability in the form of the so-called hysteresis loop. On the basis of this
concept the time series of global ice volume over the past 1000 kyr, which is in good agreement with the
time series of δ18O concentration in the sea sediments, is obtained by the model bifurcation equation, (iii)
principle of the structural invariance of balance equations of climate models evolving on the different
time scales is proposed. Such invariance, which is the direct indicator of correctly "guessed" physics of
processes on different time scales, at the same time sets the unambiguous rules of transition from one
time scale to other, for example, the transition from one-zonal model of the Earth’s global climate on
millennial time scale to multi-zonal model of Earth’s weather on the shot time scale, (iiii) also, the socalled "CO2 doubling " problem is discussed.
____________________________________________________________
1
Corresponding author: Prof. Rusov V.D., Head of Department of Theoretical and Experimental Nuclear
Physics, Odessa National Polytechnic University, Shevchenko av. 1, Odessa, 65044, Ukraine
Fax: + 350 482 641 672, E-mail: [email protected]
1. Introduction
In Ref. [1] was shown that the basic equation of energy-balance model of the Earth global
climate is the bifurcation equation (with respect to the Earth surface temperature) of fold
catastrophe with two governing parameters, i.e. insolation variations and the Earth magnetic
field variations (or cosmic ray intensity in the atmosphere). A general bifurcation problem
following from the energy-balance model of the Earth global climate (see Eqs. (20)-(23) and
(26)-(28) in Ref. [1]) for determination of global temperature Т(t) and its increment ∆Т(t) is
reduced to finding of stable solution set of equations (see Eqs. (29)-(30) in Ref.[1]):
∂
U ∗ (T , t ) = Tt 3 + a (t ) ⋅ Tt + b(t ) = 0 ,
∂T
(1)
∂
~ (t ) ⋅ ∆ T + b~ (t ) = 0 ,
∆ U ∗ (∆ T , t ) ≅ ∆ Tt 3 + a
t
∂T
(2)
where all designations are the same as in Ref. [1].
A purpose of this paper is a comparative analysis of solution of Eqs.(1)-(2) of the energybalance model of the Earth global climate [1] and well-known experimental time series of
palaeotemperature (e.g., the Vostok ice core data [25] over the past 420 kyr and the EPICA ice
core data [24] over the past 730 kyr) as well as the theoretical time series of climatic sensitivity
of the water (liquid and vapour) in the atmosphere calculated by Eqs. (1)-(2) and the time series
of δ18O isotopic concentrations (conditional analogue of ice volume) measured in the wellknown deep-water experiments.
2. Hypothesis of the North substrate and the response function of ice sheets
Now, when everything is ready for the searching the solution of the system of the initial
equation (1) and the “perturbed” equation (2) (with consideration of the initial and boundary
conditions), several important remarks have to be made about the specificity of the perturbation
Eq. (2).
It is well known that “ice sheets influence climate because they are among the largest
topographic features on the planet, create some of the largest regional anomalies in albedo and
radiation balance, and represent the largest readily exchangeable reservoir of fresh water on
Earth” [2]. Moreover, recently the Northern Hemisphere ice sheets influence on the global
climate change is discussed within the framework of the hypothesis of the so-called Northern’s
substrate effect, which explains the mechanism of nearly synchronize the climate of the Northern
and Southern Hemispheres at orbital time scales despite asynchronous insolation forcing in these
Hemispheres. The essence of the hypothesis consists in the fact that the “effect of the substrate
underlying the Northern Hemisphere ice sheets is to modulate ice-sheet response to insolation
forcing” [2].
In order to take into account the influence of the Northern’s substrate in our equations, it
is necessary to consider such variations of insolation, which influence directly the value of icesheet response [2]. At the same time, the basic equation of state (1) comprises the average annual
on the eccentricity of the Earth e(t). Therefore its variations can not be the cause for non-linear
ice-sheet response. On the other hand, the perturbed equation of state (2) contains the entropy
term ∂∆Gw+v/∂(∆T), which characterizes by definition (see Eqs. (16)-(17) in Ref. [1]) the mass
increment of water and vapour in the atmosphere ∆mw+v in the following way:
~
~ 2a µ Tt + bµ 
∂
 H ⊕ (t ) ~ ∆ m w+ v ,
∆ G w+ v ( ∆ T , t ) =  a
0 ∆ T + b0
∂ (∆ T )
2


(3)
where ∆mw+v =〈ρ〉(awv+bwv∆Т)H⊕(t); 〈ρ〉 is averaged density of the mixed water (vapour and
liquid) in the atmosphere. Let us remind that H⊕=Mtχt=0/ Mt=0χt (see Eq.(15) in [1]). Necessary
experimental data of magnetization Mt and magnetic susceptibility χt on the millennial time
scales for period of time t ∈ [0,2.25] million years are presented in Ref. [3]
It is obvious that the variations of the additional mass of water and vapour in the
atmosphere ∆mw+v within the framework of our model are in essence the variations of the
evaporated fresh water on Earth. In this context two reasons have to be mentioned, which had an
influence on the further selection of the procedure of perturbation simulation in Eq. (2). On the
one hand, it is known that according to the Northern’s substrate hypothesis just the Northern
Hemisphere ice volume is considered as a reservoir of fresh water on Earth, which plays one of
the major roles in the mechanism of the powerful ice-sheet response to the insolation forcing. On
the other hand, it is known that the auroral zones or, in other words, the zones of permeation of
the cosmic rays with the highest intensity in the Earth’s magnetosphere are in the vicinity of the
65o Northern and Southern latitude. Moreover, the ice formed in latitude 65 oN has a “lifetime”,
which is significantly longer than in the analogous latitude in the Southern hemisphere, since it is
formed for the most part on the terrestrial surface of the continents in the Northern hemisphere,
and not on the water surface of the southern oceans, which, moreover, are heated in these
latitudes (~65oS) by the global warm stream. In addition, the maximum variations of insolation
are often observed exactly in the vinicity of the 65o northern and southern latitudes because the
continental ice formed in the auroral zone region of the Northern hemisphere plays the role of a
super amplifier of the insolation variations in at 65oN, while the floating ice in the region ~65oS
has a low response to high insolation variations in 65oS due to its insignificant volume.
Obviously, this is the reason for using long-term variations of June insolation at 65 oN in the
simulation of Northern Hemisphere ice volume [4].
Hence, for the emulation of the Northern’s substrate effect in our model, it was necessary
to perform such transformations in Eq. (2) (in the form of the average annual insolation
“perturbation” of the Earth <W0> ≅S0/4), as a result of which the variations of water and vapour
mass in the atmosphere ∆mw+v could be modulated not only by the Earth’s magnetic field
variations, as seen from Eq. (3), but also by the variations of June insolation at 65 oN (via the
temperature variations).
For this purpose, let us write down a balance equation (analogous to Eq.(25) from Ref.
[1]) but for the sum of the normalized variations of insolation ∆Wi for all i-latitudes at the time
moment j:
1 + 90
1 + 90
1 + 90
γ
W
=
γ
W
+
∑ i ij γ ∑  i ij γ ∑ γ i ∆ Wijσ
γ i = − 90
i = − 90
i = − 90



ij
,
(4)
1
tn
tn
∑
j= 0
Wij = Wij ,
where γi /γ is the part of the surface corresponding to i-latitude; −90°≤ i ≤ 90°; tn is the length of
the sampling in time; σij is standard error of the current insolation value Wij with respect to the
average <Wij> in i-latitude at the time moment j∈[0,tn]. Note that at tn=1000 kyr the number of
sampling discretizations n, for example, is usually equal to 100.
Now we introduce so-called response function of the continental and floating ice of the
Earth h(i), which characterizes the Earth ice-sheet response to insolation forcing [2]. If the ice
volume in 65oS latitude (Antarctic peninsula) is neglected, the response function h(i) may be
written in the most simple manner as follows
h(i) =
(5)
γ
γ
65
W0
W65
δ (i − 65  ) +
γ W
∑  γ W0 δ (i − k ) +
k = 64
k
k
+ 90 
γ W0
δ (i − k ) ,
k = − 64  γ k Wk
− 90
∑
where −90°≤ i ≤ +90°;
S0
S
≅ 0
2
4
4(1 − e )
W0 =
(6)
is by definition the average annual insolation of the Earth, which very weakly depends on the
eccentricity е(t) of the Earth's elliptic orbit.
It is clear, that introduced in this manner Eq. (5) emulates the anomalous behaviour of the
response function of continental and floating ice of the Earth. Then the “perturbation” of the
average annual insolation of the Earth <W0> caused by the effect of the response function (5)
may be obtained by multiplication of the function (5) by Eq.(4). Moreover, if take into account
that
∆ Wi σ
i
≅ − ∆ W− iσ
−i
,
(7)
the effect of response function (5) to Eq. (4) results into following equation:
W0
W65
W65
= W0 +
W0
∆ W65σ
W65
65
.
(8)
Apparemment, this is the formal mathematic presentation of the physical essence of the
so-called Northern substrate [2].
The expression for the reduced average annual insolation of the Earth (8) we rewrite in a
more simple and convenient form
4Wreduced (t ) = S 0 + ∆ Wˆ σ
S
,
(9)
where
Wreduced (t ) ≡ W65
W0
W65
,
W0 ≅
S0
,
4
∆ Wˆ ≡ ∆ W65 , σ
S
≡
4 W0
W65
σ
65
.
(10)
At the same time the reduced standard error looks like
σ
S
=
4 W0
W65
σ
65
=
4 W0
k W65L
σ
65
= 53.45 ,
(11)
where <W0>= S0/4 is the average annual insolation of the Earth; S0=1366.2 W/m2 is the solar
L
constant, which taken from Ref. [5]; calculated data W65
at k=S0/ S 0L =1366,2/1350=1,012
L
from Ref. [6] are used as values for the average insolation <W65>=k W65 =k437,457 W/m2) and
standard error σ65=17,321 in 650N latitude.
Generally speaking, for the formation of the universal response function of the Earth’s
surface the complete information on the spatial-temporal distribution of ice (fresh water) on the
Earth’s surface has to be taken into account. However, as seen from the numerical experiment
considered below, it turned out that as first approximation it is sufficient to use the response
function of the Earth’s surface in the form (5).
4. Numerical experiment
It is obvious, to solve the bifurcation Eqs. (1)-2), which describes extreme values of
temperature T and increment ∆T, it is necessary to determine the values of three climatic
constants aµ , bµ and β in Eqs. (27)-(28) from Ref. [1]; the values of other constants are usually
known.
The value of β was determined from known estimation of the power of heat energy reradiated by carbon dioxide, i.e. WCO2 ~1.7 W/m2 [7], in which the sum of re-emission energy
variations ∆ GCO (Tt = 0 ) in Eq. (1) and ∆ GCO ( ∆ Tt = 0 ) in Eq.(2) as linear approximation
2
2
satisfies following equality:
[
]
1
1
1
1
∆ GCO2 (Tt = 0 ) + ∆ GCO2 (∆ Tt = 0 ) = β Tt = 0 = WCO2 = ⋅ 1.7
γ
2
2
2
[W
]
m2 ,
(12)
where H⊕(t=0)=1; the coefficient equal to 1/2 reflects the re-emission isotropy of carbon dioxide.
Below the initial condition of the modern climatic representative temperature T0 [8] will be taken
into account:
T0 = Tt = 0 + ∆ Tt = 0 = 288.6 ± 1.0
(13)
Taking into account Eqs. (1) and (13), we obtain from Eq.(12):
[K ] .
β = 1.7 Tt = 0 = 0.006
[W
]
m2 K .
(14)
As a result of simulation of the solution of the system of the initial equation (1) and
“perturbed” equation (2) with allowance for the initial conditions (13)-(14) the values of climatic
constants
a µ = 0.222,
bµ = − 127.249
(15)
and the final values of initial parameters of the system of equations (1), (2):
Tt = 0 = 286.031
[ K ],
∆ Tt = 0 = 2.100
[ K ],
β = 0.006
[W
]
m2 K ,
(16)
which satisfy the degree of uncertainty of modern climatic temperature T0 described by Eq. (13).
Note, that the investigation of the parameterized solution of the system of equation (1)(2) (i.e., temperatures Tt+∆Tt) exhibits high degree of solution stability with respect to relatively
small “perturbations” of the initial and boundary parameters (15)-(16). In other words, the small
variations of the climatic constants aµ and bµ, lead to small changes of the initial parameters Tt=0,
∆Tt=0 and β. Running a few steps forward, it is important to note that the general solution of the
equation system (1)-(2) or, more exactly, the time distribution of theoretical palaeotemperatures
Tt+∆Tt, according to the numerical experiment practically does not change its form at small
variations of the climatic constants (15) and initial parameters (16). This means that the
approximate invariance of the form of time distribution of theoretical temperature Tt+∆Tt is
predetermined by some attractor in the phase portrait of the physical system (1)-(2) and needs a
special physical-mathematical substantiation, but this is already another problem, going beyond
the scope of the present work.
Let us consider the solution of Eq. (1) with respect to the temperature Tt obtained with
allowance for the parameters (15)-(16). First, the dependence of temperature Tt on the relative
magnetic field H⊕ (Fig.1а) is considered. From Fig. 1a it follows that the temperature Tt is very
low when H⊕ ≤0.5. The existence of such evidently non-physical range of the temperatures Tt is a
reflection of the rough approximation used in deriving Eq. (14) in Ref. [1]. In the view of
physics, it apparently means that Eq.(14) in Ref. [1] does not take into account that the influence
of solar wind on the galactic cosmic ray flux and Earth’s magnetosphere by reconnection has a
threshold character. In other words, formula (14) does not take into account the possibility of
existence of such a minimum value of solar wind magnetic induction, when the intensity of
galactic cosmic ray “sweep” and interaction of solar wind with Earth’s magnetosphere are
although “minimal” but are nonzero. Moreover, this weak interaction between solar wind and
Earth’s magnetosphere is veiled by the eigen variations of the Earth’s magnetic field, which are
determined by the specificities of the Earth’s liquid core internal kinetics and by the stochastic
properties of the magnetic field energy source, i.e. slow nuclear burning on the boundary of
liquid and solid phases of the Earth’s core (a georeactor of 30 TW) [9].
On the other hand, it is clear that the dependence of temperature Tt on the relative
magnetic field H⊕ (Fig. 1а) by virtue of Eq.(1) type has to be of a hysteresis form. For this
reason, in the range of H⊕ ≤0.5 the real curve in Fig. 1a has to be with several Kelvins lower (for
example 3-5 K) than the part of the curve located in the H⊕ ≥0.5 range. The subsequent
numerical experiments on finding the solution) of the equation system (1)-(2), i.e. temperatures
Tt+∆Tt, show that at H⊕ ≤0,5 the “basic” temperature TH
TH
⊕
≤ 0.5
= 282
⊕
≤ 0.5
[K],
of ECS (Fig. 1b)
when
H ⊕ ≤ 0.5
(17)
is the best value from the viewpoint of the fitting procedure of experimental data from the
Vostok ice core [11] and from the EPICA ice core [12].
Fig. 2d shows graphically the numerical solution of Eqs. (1)-(2), which describes time
evolution of temperature Tt+∆Tt with respect to the average temperature Tt=0=286 K. To compare
our results with experimental data, Fig. 2e shows the palaeotemperature data over the past 420
kyr from the Vostok ice core [11] and over 730 kyr from the EPICA ice core [12].
Thus the high goodness of fit between the experimental (Fig.2e) and theoretical (Fig.2d)
data is a validity indicator of the main assumption used in our model; we assume that the
temperature of ECS is caused by both the insolation variations and GCR intensity variations (or,
equivalently, the Earth magnetic field variations).
However, the problem of the existence of so-called anomalous palaeotemperature jumps
has to be mentioned. Time location of these temperature jumps are shown by red vertical lines in
Fig. 2e. As anomalous palaeotemperature jumps we imply sharp temperature rise, which, as a
rule, occurs at once after synchronous passing a minimum by insolation (see Fig.2b) and the
Earth’s relative magnetic field (see Fig. 2c). Below we attempt to find the way of solving such a
physical problem and thereby explain, why such jumps are not described within the frame of the
proposed bifurcation model of global climate.
Comparing the time series of insolation, relative magnetic field and experimental
palaeotemperature (Fig. 2), it is possible to conclude that neither insolation nor the Earth’s
magnetic field variations can be such energy source, which generates these temperature
anomalies. By virtue of energy conservation law, an additional pulsed heat source must to exist
in the energy balance of ECS (see Eq.9 and also Fig. 5 in Ref. [1]) in order to explain this heat
paradox. Let us consider the physical mechanism, which, in our opinion, predetermines the
operation of such a source.
It is known that solar wind magnetic activity force on the Earth’s magnetosphere on the
millennial time scale is modulated by the Earth’s elliptic orbit eccentricity. The changes of the
interplanetary magnetic field induced by the solar wind, when higher than a certain threshold, by
virtue of the law of electromagnetic induction may become the reason of inductive charged
current production in the Earth liquid core. On other words, when orbital eccentricity approaches
to its local maximum, the variations of magnetic flux of interplanetary field can generate
substantial variations of the Earth magnetic field, which, in its turn, will can be induce a charged
current in the Earth liquid core. The such additional charged current, which by virtue of Le
Chatelie law has opposite direction relative to moving direction of convection current in the
Earth liquid core, can partly block, thereby, destroy the convection mechanism responsible for
Earth magnetic field.
The convection partial blocking, on the one hand, will lead to sharp decrease of the Earth
liquid core rotation speed and, on the other hand, will be the reason of the strong violation of the
heat abstraction from the interface of the liquid and solid phase of the Earth core. This can
results in a significant increase of temperature in this region, where, as is known, the georeactor
of 30 TW is operating [9].
Now the consequences of these violations will be considered. First let us consider the
consequences for ECS, which will be provoked by the sharp decrease of the Earth liquid core
rotation speed. It is known that according to the angular moment conservation law, the angular
velocities of the mantle ωmuntle , the liquid ωliq and solid ωsolid phases of the Earth are related in the
following manner
I m ω muntle + I l ω liquid + I s ω solid = const
,
(18)
where Im, Il and Is are the inertial moments of the mantle, liquid and solid phases of the Earth
core, respectively.
It is clear that in conformity with (18) the decrease in the angular velocity of the Earth
liquid core will lead to increase in the angular velocity not only of the solid core but also of the
mantle rotation speed. This means that the sharp change of the mantle rotation speed will cause
strong friction between the lithosphere and the atmosphere, and this, in its turn, will lead to a
sharp increase of the average surface and tropospheric temperature of the Earth. In this way such
a physical mechanism of Solar-Earth relations, in our opinion, may be responsible for these
sufficiently rare in time (~100 kyr) but powerful heat impulses (Fig. 2e) disposable in the
atmosphere from the Earth’s surface.
Now the geophysical consequences induced by the significant increase of the Earth liquid
core temperature will be considered. It is known [9] that the induced fission cross-section of 238U,
which is the natural fuel of the georeactor [9], strongly depends on the medium temperature Т (σf
~T4). It means that the nuclear burning rate on the boundary of liquid and solid phases of the
Earth will increase with medium temperature growth. Moreover, the nuclear burning rate will
continue increasing until the temperature at the liquid-solid interphase of the Earth core is raised
significantly above the average temperature of the liquid core that, in its turn, will lead to the
recovery of the ordered convection in the liquid core. In addition, the direction of the recovered
convection may or may not coincide with the direction of convection in the liquid core prior to
its destruction. In the case of non-coinciding directions of the recovered and disturbed
convection, a very interesting effect is observed, i.e. so-called the Earth magnetic field inversion.
Obviously, that the average period of such inversions will correlates with period of the Earth
orbit eccentricity. This agrees with conclusions of Ref. [13] devoted to the research of stochastic
mechanism of geomagnetic poles inversions. Thus, in case of abrupt rising of the Earth liquid
core temperature, the georeactor plays the role of an energy stabilizer of ordered convection in
the Earth liquid core and the Earth magnetic field, respectively.
5. Discussion and conclusions
To analyze the behaviour of a complex climatic systems, it is convenient to use the notion
of so-called climatic sensitivity, e.g. in the sense of the definition proposed by Wigley and Raper
[14]. In our case the climatic sensitivityλw+v with consideration of Eq.(1) (see Eq. (31) in Ref.[1])
defines a rate of total water mass ∆mw+v (vapour and liquid) change in the atmosphere with
respect to the temperature increment ∆Т:
λ w+ v = −
1
∂
1
∆ G w+ v (∆ T , t ) ~
∆ m w+ v ,
∆ t ∂ (∆ T )
∆t
(19)
where ∆t =10 kyr is time scale resolution (see Fig. 2).
The physical sense of this value is clearly shown in Fig.3, from which the climatic
sensitivity is an adequate valuation of temperature bistability intrinsic to the "potential" of fold
catastrophe (see Eqs.(20) and (26) in Ref.[1]). Also, the decrease (increase) of increment rate of
total water mass (vapour and liquid) in the atmosphere λw+v predetermines, by virtue of
conservation law, the same by value increase (decrease) of increment rate of total fresh water
mass ∆Ffwf in the ECS. This shows that there is an interrelation between the global temperature of
ECS and rate of fresh water mass change (Fig.2), which evolution is provided by the global
ocean circulation [15]. Moreover, such an interrelation can also display the temperature
instability in the form of so-called hysteresis loop analogous to that shown in Fig.3.
Thus, along with the parameter λw+v, the parameter ∆Ffwf f determining the rate of change
of fresh water mass with respect to the global temperature is also climate sensitive parameter.
Generally speaking, it can be supposed that all physical, chemical, and biological parameters
displaying the temperature instability in the form of the so-called hysteresis loop belong to
variable climate-sensitive parameter class, but generally on different geological time-scale. It is
in turn obvious that the climate-sensitive parameters attached to particular geological time-scale
must be linearly connected with each other. This statement can be easily shown (Fig.4) by the
example of series of linearly-connected parameters:
− λ w + v ~ ∆ F fwf ~ I V
,
(20)
where IV is the global ice volume.
Moreover, the evident correlation (see Fig.4) between the known time series of δ18O in
the sea sedimentary (ice volume proxy) [16,17] and analogous time series of climatic sensitivity
calculated by Eqs. (19) and (3) indicates both the existence of direct linear relationship between
these parameters and forecasting ability of the discussed bifurcation model of the Earth global
climate (see Figs. 2 and 4).
On the other hand, it is known that the temperature instability of global climate like other
analogous instabilities of the wide class of bistable systems can cause the mode of stochastic
resonance (SR) under certain conditions [18,19]. For example, using an ocean-atmosphere
climate model (which belongs to bistable models), Ganopolski and Rahmstorf [45] ascertained
that the stochastic resonance is the main mechanism of climate variability on secular time scale
during last glacial period. Also, the modeled warm events agree in many respects with the
properties of so called the Dansgaard-Oeschger events recorded by isotope measurements of
δ18O in Greenland ice [20-22]. This is important outcome, which moreover agrees very well with
other experimental observations. Because it is interestingly to show that this outcome is easily
described by the proposed bifurcation model of Earth climate.
The stochastic resonance consists in the possibility of the stochastic system tuning (for
example, ECS) by varying the noise intensity for the modulation signal maximal multiplication
[18]. The systems displaying the stochastic resonance belong to nonlinear systems with the
characteristic dynamics, which is in a certain sense intermediate between the regular and chaotic
dynamics. In the simplified form the mechanism of stochastic resonance can be explained in
terms of particle Brownian motion in the bistable potential under the action, for example, of
white noise in the presence of weak periodic signal.
In our case, the Earth climate system lying in asymmetric bistable potential (see Eq.(26)
in Ref. [1]) can be considered as a Brownian particle. "Glacial" time scale, on which an
appearance of warm DO-events can be expected, is in the 20-75 kyr interval. On the one hand,
this theoretical interval coincides with the analogous experimental interval for the actual DOevents obtained by NGRIP collaboration using the high-resolution (∆t=50 yr) method of δ18O
concentration measurement, which is the effective indicator of global ice volume [22]. On the
other hand, supposing that coefficients of bistable potential are constant in this interval, Eq.(26)
from Ref.[1] looks like
∆ U (∆ T , t ) =
~
1
1~
2
∆T4 − a
− bDO ∆ T ,
DO ∆ T
4
2
~ , b~ > 0 .
a
DO
DO
(21)
As was shown in Ref. [21], the insolation S(t) on secular time scale is under the combined
action of two harmonic motions with amplitudes A1 and A2 induced by so-called DeVries-Suess
[23] and Gleisberg [24] solar cycles with periods T1= 210 years and T2= 86.5 years
S (t ) = S 0 + Ω (t ) = S 0 + A1 sin(ω 1t + ϕ 1 ) + A2 sin(ω 2 t + ϕ 2 ) ,
(22)
where ω1=2π/T and ω2=2π/T2 are frequencies of this cycles.
As it was shown experimentally and theoretically in Refs. [20,21], superposition of these
two frequencies generates the resulting oscillation with period of 1470 years. Let us remind that
in this case insolation term in Eq. (28) from Ref. [1] on the secular time scale has the form:
4Wreduced (t ) = S 0 + Ω (t ) + ∆ Wˆ (t )σ
S
.
(23)
On the other hand, the high time scale resolution (∆t=50 yr) of the isotope data in the
NGRIP-experiment means that according to the known behavior of temperature spectral density
of ECS (see Fig. 3) in Ref.[1]) the action of the mixture of white noise and 1/f- noise is the main
reason of ECS temperature variations on this time interval. Then under action of weak harmonic
signal, for example, of A1sin(ω1t+ϕ1)+ A2sin(ω2t+ϕ2) type, the dependence ∆T(t) can be obtained
for over-damped system (in whose equation of motion we ignore the inertia term m x in
comparison with friction x ) using the equation of Langevin type:
x = − ∂ (∆ U ∗ ) x ( x, t ) + ζ (t ) ,
x (t ) ≡ ∆ T (t )
(24)
or with allowance for Eq. (23) and Eqs. (26)-(28) from Ref.[1]
~ x − x 3 + b~ + b~ η [ A sin(ω t + ϕ ) + A sin(ω t + ϕ )] + ζ (t ) ,
x = a
DO
DO
0 α
1
1
1
2
2
2
(25)
where x is force of friction; ζ (t)=ξ(t) + η(t) is the additive mixture of white and 1/f-noise ,
~
respectively. In case when the noise is high, we can ignore the term bDO causing the potential
asymmetry in Eqs. (21) and (25).
In scientific literature the analogue of this equation (25) is the well studied canonical
equation describing the stochastic resonance in case of particle Brownian motion in symmetric
bistable potential , which with our designations has the following form
~ x − x 3 + A sin ω t + ξ (t ) ,
x = a
DO
x(t ) ≡ ∆ T (t ) ,
(26)
where the following restriction Axm << ∆(∆U∗(x,t)) = ∆U∗(0) − ∆U∗(xm) is imposed on the
~ )
amplitude of weak external periodic signal; ± xm= ( a
DO
1/ 2
is the coordinate of potential
~2 4
minimum ∆U∗; ∆(∆U∗(x,t))= a
is potential barrier height; ξ(t) is white noise of Gaussian
DO
type with zero mean (〈ξ(t)〉=0) and correlation function of 〈ξ(t), ξ(0)〉=2Dδ(t)〉 type.
In most cases, it is convenient to identify the SR effect using an experimental
investigation of the residual-time statistics for the considered system in the potential well or,
more precisely, using the so-called residence-time distribution [26]
P (t ) ~ exp( − W K t ) ,
(27)
WK =
1
τ
K
1
=
2π
 d 2 ( ∆ U ∗ )

dx 2

x= 0
d 2 (∆ U ∗ )
dx 2
(28)


x = xm 

12
 ∆ ( ∆ U ∗ ) ± x m A sin ω t 
 ,
exp −
D


where WK and τK are the modified Kramer’s rate and Kramer’s time respectively [44]. These
values define the rate and time for the transition of an excited system through a potential barrier
under the conditions of weak periodic perturbation.
It is obvious that in the construction of residence-time distribution function (27) due to
the existence of the periodic component in Eq. (28) a series of peaks corresponding to odd
harmonics centered on times multiple of the steering command half-cycle
Tn ≅ (2n + 1)
T
π
= (2n + 1) ω ,
ω
2
n = 0,1,... ,
(29)
where ω and Tω are the frequency and period of external recurring perturbation, respectively.
Moreover, it follows from analysis of Eq. (28) that the peaks (29) appear only when the
weak harmonic perturbation is included; in the opposite case the monotone exponential decay of
function (27) is observed. Thus the physical interpretation of Eq. (27) is obvious enough and lies
in the fact that an exponential background is the part of the residence-time distribution function
(27), which is produced by the changeovers caused by the noise only, whereas the peaks (29) are
the reflections of synchronization between the escape mechanisms and external periodic
perturbation.
Just using the ideology of optimal synchronization in the simulation of stochastic
resonance equation of Eq. (26) type in the framework of the bistable climate model made is
possible to obtain the "experimental" function of the residence-time distribution [20, 21].
Moreover, the properties of the modeled warm events, using which the residence-time
distribution was obtained, are in good agreement with the real properties of the DansgaardOeschger events recorded by δ18O isotope measurements in Greenland ice [22].
It is interestingly to note, that comparing the stochastic Eq. (25) corresponding to the
secular time scale and bifurcation equations of global climate (1) and (2) constructed on the
millennial time scale, we can see that they have the identical structural terms of (ax−x3+b) type
on right-hand side. As is shown in Ref. [1], in the case of the correct "guessing" of physical
essence of process the appearance of identical model structural terms on the different time scales
is key property of hierarchical models, which is expressed as of structural invariance principle of
balance equations evolving on the different time scales [1]. In other words, in our case this
property consists in that climate balance equations save a self-similar macroscopic structure, for
example, of (ax−x3+b) type and its governing parameters (a, b) in any ECS models
independently of used time scale.
In this sense, it is appropriate remind a known model of oscillator with the "retarded
argument" [27], which enough realistically reflects a nonlinear dynamics of temperature
oscillations (the ∆T anomalies) of the EL Nino current on the annual time scale and has the
following modified form:
x = ax − x 3 + b − µ ⋅ x(t − ∆ ) + Υ (t ) + ℜ + ξ (t ) ,
x(t ) ≡ ∆ T (t ) ,
(30)
where the first term on right-hand side of Eq. (30) describes the strong positive feed-back with
the framework of the ocean-atmosphere system; the second term bounds the growth of unstable
perturbations of ocean temperature; b is constant parameter; the fourth term emulates a motion
process of equatorial "trapped" waves (Rossby "latent" waves) through the Pacific ocean, which
after the elastic reflection on the western continental boundary return (as the Kelvin waves)
along equator to the central part of the Pacific ocean with time delay ∆; a fifth term is Υ(t)
function, characterizing the continuous changes of ocean surface average temperature during
year (Υ(t)); sixth term (ℜ≈5°С/100 years) emulates a process of global warming; ξ(t) is white
noise of Gaussian type with zero mean (〈ξ(t)〉=0) and standard deviation λ [27].
It is obviously, that the considered hieratic chain of the ECS climatic states, which
consists from oscillation model of the AL Nino temperature anomalies (on the annual time scale)
⇒ stochastic model of heat DO-anomalies during the last glacial period (on the secular time
scale) ⇒ bifurcation model of global climate (on the millennial time scale) really contains an
invariant macroscopic structure of (ax−x3+b) type, that is direct corollary of structural invariance
principle of the balance equations of ECS on the different time scales. Moreover, it is necessary
to add that of structural invariance principle is not only the direct indicator of correctly guessed
physics of process on different time scales but it simultaneously sets obvious rules of transition
from one time scale to other.
Finally, let us adduce the short comment for the so-called "CO 2 doubling" problem. A
~
simple analysis of structure parts in the time-dependent governing parameter b (t) denotes the
fact that it contains the three independent components (see Eq. (28) in Ref.[1]), i.e. the insolation
variations, re-emission energy of water total mass (vapour and liquid) in the atmosphere and the
climatic constant β defining the degree of variability of the differential heat power reradiated by
atmospheric carbon dioxide. Also, using Eq. (28) from Ref. [1] it is easy to show that the
following inequality
1
1
β < < η α Wreduced (t ) − 4δ σ Tt 3 + ( 2a µ Tt + bµ ) H ⊕ (t ) ,
2
2
for
∀ t ∈ [0,730kyr ]
(31)
practically always fulfills for any t=0÷730 kyr.
Inequality (31) implies that considerable influence of anthropogenic perturbation is only
possible when the differential heat power reradiated by atmospheric carbon dioxide βperturb is
appreciably increased in comparison with the present-day value β (see Eq.(14)). For example,
using evident modification of Eq. (28) from Ref.[1 it is easy to show by computational
experiment that on the interval t=0÷120 kyr the threshold anthropogenic thermal effect occurs
under the 100-time decrease of the present-day value β. In other words, in the framework of the
discussed bifurcation model of the Earth global climate the so-called anthropogenic "CO 2
doubling" problem is practically absent.
It can be concluded from the above mentioned that, in our opinion, the most important
statement of the presented model is the fact that the Earth global climate, on the one hand, is
completely determined by the two governing parameters, i.e. by insolation and galactic cosmic
rays, and, on the other hand, it practically has not restrictions on global forecast horizon on the
millennial time scale, if theoretical or experimental time series of Earth magnetizing force H⊕(t)
are known, i.e. it is quite predictable.
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FIGURE CAPTIONS
Fig. 1. Theoretical dependence of temperature Tt on relative magnetic field H⊕ (solid
curve) obtained from Eq.(1) (see Eq. (29) in Ref. [1]) with allowance for Eqs.(21)-(22) from Ref.
[1]. In computations the dependence, which has hysteresis form (dotted curve) and meets a
condition (17), were used. Explanation is in text.
Fig. 2. Model of ECS temperature response to insolation variations and Earth magnetic
field variations on the 1000 kyr time domain. In figure is shown time evolution of (a) Earth
orbital eccentricity [5], (b) insolation variations in latitude 65°N (summer) [5], (c) variations of
Earth magnetic paleointensity [3], (d) variations of Earth surface model temperature (with
respect to average today’s temperature Tt=0=286.0 K) obtained by solving the system of equations
(1)-(2), (e) variations of deuterium isotopic concentration δD (conditional analogue of
paleotemperature) measured in the ice cores within the framework of the Antarctic projects
"Vostok" on the time domain 420 kyr [11] and EPICA Dom C on the time domain 740 kyr [12].
The anomalous jumps of variations of deuterium isotopic concentration δD are indicated by
arrows. The vertical red lines indicate time of possible anomalous temperature jumps and,
consequently, variations of deuterium isotopic concentration δD.
Fig. 3. Dependence of variations of theoretical increment of temperature () on the
climatic sensitivity λw+v calculated by Eqs. (19) and (3). Insert - known hysteresis dependence of
temperature on rate of increment of fresh water total mass in the high-latitudinal experiments in
North Atlantic [28].
Fig.4. Comparison of time series of isotopic concentrations of δ18O (conditional analogue
of ice volume) measured in the deep-water experiments: (a) Bassinot et al. [16] (solid blue line)
and Imbrie at al. [16] (dashed red line), and (с) Tidemann et al. [17] and (b) theoretical time
series of climatic sensitivity λw+v calculated by Eqs. (3) and (19).
Fig. 1
Fig. 2
Fig. 3
Fig.4