On the critical mass of greenhouse gas
In recent years, changes in the Earth’s climate have raised concern all around the globe. Climatologists have been drawing connections between global warming and a growing number of natural disasters, unexpected temperature fluctuations in some regions of the world and a number of other climatic abe...
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Інститут геофізики ім. С.I. Субботіна НАН України
2018
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irk-123456789-1326572018-04-25T03:02:52Z On the critical mass of greenhouse gas Klimchuk, E.F. Tarasov, V.F. In recent years, changes in the Earth’s climate have raised concern all around the globe. Climatologists have been drawing connections between global warming and a growing number of natural disasters, unexpected temperature fluctuations in some regions of the world and a number of other climatic aberrations. Within the scientific community, the opinions as to the nature and mechanism of the Earth’s climate change have split. Some contend that since the beginning of the industrial revolution, the carbon dioxide levels in the air have been steadily rising due to human production activities. Along with other gases, carbon dioxide has been inculpated for the greenhouse effect. In connection with this, a number of recent international conferences have adopted resolutions to reduce carbon dioxide emissions into the atmosphere. Another group of climatologists bases its findings on observations of solar activity, arguing that global warming is caused by a recurring spike in solar activity, with the current increase due to end soon, potentially giving way to a new ice age down the road. In the following work, we put forth yet another hypothesis regarding global warming. The influence of four main positive feedback loops caused by the secondary emission of water vapor, ÑÎ2, ÑÎ4, and decreased albedo on the earth climate system is shown on the basis of the general theory of feedback. If the present level of primary anthropogenic emissions of greenhouse gas (GhG) keeps, the total mass of atmospheric greenhouse gas can run up to such a critical value that the mentioned feedbacks, which give rise to self-amplification of the greenhouse effect, can cause the bifurcation transition of the climate system to the state of self-heating tending to the unlimited rise of mean temperature of the earth surface. В останні роки змінення клімату Землі непокоїть громадськість світу. Зростання стихійних лих, раптові температурні коливання в окремих регіонах світу та інші відхилення клімату від традиційної поведінки кліматологи пов'язують з глобальним потеплінням. Наукова громадськість з кліматології розділилася щодо питання механізму природи потепління клімату Землі. Одні дослідники вважають, що з моменту початку промислової революції виробнича діяльність людства з кожним роком збільшує кількість вуглекислого газу у повітрі. Як установлено, поряд з іншими газами він відповідає за парниковий ефект. У зв'язку з цим останнім часом відбулося кілька міжнародних конференцій, на яких ухвалено рішення про скорочення викидів вуглекислого газу в атмосферу. 2018 Article On the critical mass of greenhouse gas / E.F. Klimchuk, V.F. Tarasov // Геофизический журнал. — 2018. — Т. 40, № 1. — С. 70-77. — Бібліогр.: 12 назв. — англ. 0203-3100 DOI: 10.24028/gzh.0203-3100.v40i1.2018.124016 http://dspace.nbuv.gov.ua/handle/123456789/132657 551.583.2 en Геофизический журнал Інститут геофізики ім. С.I. Субботіна НАН України |
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In recent years, changes in the Earth’s climate have raised concern all around the globe. Climatologists have been drawing connections between global warming and a growing number of natural disasters, unexpected temperature fluctuations in some regions of the world and a number of other climatic aberrations. Within the scientific community, the opinions as to the nature and mechanism of the Earth’s climate change have split. Some contend that since the beginning of the industrial revolution, the carbon dioxide levels in the air have been steadily rising due to human production activities. Along with other gases, carbon dioxide has been inculpated for the greenhouse effect. In connection with this, a number of recent international conferences have adopted resolutions to reduce carbon dioxide emissions into the atmosphere. Another group of climatologists bases its findings on observations of solar activity, arguing that global warming is caused by a recurring spike in solar activity, with the current increase due to end soon, potentially giving way to a new ice age down the road. In the following work, we put forth yet another hypothesis regarding global warming. The influence of four main positive feedback loops caused by the secondary emission of water vapor, ÑÎ2, ÑÎ4, and decreased albedo on the earth climate system is shown on the basis of the general theory of feedback. If the present level of primary anthropogenic emissions of greenhouse gas (GhG) keeps, the total mass of atmospheric greenhouse gas can run up to such a critical value that the mentioned feedbacks, which give rise to self-amplification of the greenhouse effect, can cause the bifurcation transition of the climate system to the state of self-heating tending to the unlimited rise of mean temperature of the earth surface. |
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Klimchuk, E.F. Tarasov, V.F. |
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Klimchuk, E.F. Tarasov, V.F. On the critical mass of greenhouse gas Геофизический журнал |
| author_facet |
Klimchuk, E.F. Tarasov, V.F. |
| author_sort |
Klimchuk, E.F. |
| title |
On the critical mass of greenhouse gas |
| title_short |
On the critical mass of greenhouse gas |
| title_full |
On the critical mass of greenhouse gas |
| title_fullStr |
On the critical mass of greenhouse gas |
| title_full_unstemmed |
On the critical mass of greenhouse gas |
| title_sort |
on the critical mass of greenhouse gas |
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Інститут геофізики ім. С.I. Субботіна НАН України |
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2018 |
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| citation_txt |
On the critical mass of greenhouse gas / E.F. Klimchuk, V.F. Tarasov // Геофизический журнал. — 2018. — Т. 40, № 1. — С. 70-77. — Бібліогр.: 12 назв. — англ. |
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| fulltext |
E. F. KLIMCHUK, V. F. TARASOV
70 Ãåîôèçè÷åñêèé æóðíàë ¹ 1, Ò. 40, 2018
ÓÄÊ 551.583.2
On the critical mass of greenhouse gas
E. F. Klimchuk , V. F. Tarasov, 2018
G. V. Kurdyumov Institute for Metal Physics,
National Academy of Sciences of Ukraine, Kiev, Ukraine
Received 1 October 2017
 îñòàíí³ ðîêè çì³íåííÿ êë³ìàòó Çåìë³ íåïîêî¿òü ãðîìàäñüê³ñòü ñâ³òó. Çðîñòàííÿ
ñòèõ³éíèõ ëèõ, ðàïòîâ³ òåìïåðàòóðí³ êîëèâàííÿ â îêðåìèõ ðåã³îíàõ ñâ³òó òà ³íø³
â³äõèëåííÿ êë³ìàòó â³ä òðàäèö³éíî¿ ïîâåä³íêè êë³ìàòîëîãè ïîâ�ÿçóþòü ç ãëîáàëüíèì
ïîòåïë³ííÿì. Íàóêîâà ãðîìàäñüê³ñòü ç êë³ìàòîëî㳿 ðîçä³ëèëàñÿ ùîäî ïèòàííÿ ìåõà-
í³çìó ïðèðîäè ïîòåïë³ííÿ êë³ìàòó Çåìë³. Îäí³ äîñë³äíèêè ââàæàþòü, ùî ç ìîìåíòó
ïî÷àòêó ïðîìèñëîâî¿ ðåâîëþö³¿ âèðîáíè÷à ä³ÿëüí³ñòü ëþäñòâà ç êîæíèì ðîêîì çá³ëü-
øóº ê³ëüê³ñòü âóãëåêèñëîãî ãàçó ó ïîâ³òð³. ßê óñòàíîâëåíî, ïîðÿä ç ³íøèìè ãàçàìè
â³í â³äïîâ³äຠçà ïàðíèêîâèé åôåêò. Ó çâ�ÿçêó ç öèì îñòàíí³ì ÷àñîì â³äáóëîñÿ ê³ëüêà
ì³æíàðîäíèõ êîíôåðåíö³é, íà ÿêèõ óõâàëåíî ð³øåííÿ ïðî ñêîðî÷åííÿ âèêèä³â âóã-
ëåêèñëîãî ãàçó â àòìîñôåðó.
²íøà ÷àñòèíà êë³ìàòîëîã³â, íà ï³äñòàâ³ ñïîñòåðåæåíü çà àòèâí³ñòþ Ñîíöÿ, ââà-
æàº, ùî ïîòåïë³ííÿ ñïðè÷èíåíî ïðîÿâîì öèêë³÷íîãî ÷àñîâîãî ïåð³îäó ö³º¿ àêòèâ-
íîñò³, ³ íåçàáàðîì öåé öèêë çàê³í÷èòüñÿ. Ïðè÷îìó â ïîäàëüøîìó ìîæëèâå íàñòàííÿ
íîâîãî ëåäíèêîâîãî ïåð³îäó. Ó ñòàòò³ çàïðîïîíîâàíî ³íøó ìîæëèâó ðåàëüíó ïðè-
÷èíó ïîòåïë³ííÿ êë³ìàòó. Íà çàñàäàõ çàãàëüíî¿ òåî𳿠çâîðîòíîãî çâ�ÿçêó ïîêàçàíî,
ùî ÷îòèðè îñíîâí³ ïîçèòèâí³ çâîðîòí³ çâ�ÿçêè, ùî âèêëèêàþòü âòîðèííó åì³ñ³þ âî-
äÿíî¿ ïàðè, CO2 , CO4 òà çìåíøåííÿ àëüáåäî, âïëèâàþòü íà êë³ìàòè÷íó ñèñòåìó Çåìë³.
ßêùî íà öåé ÷àñ ð³âåíü ïåðâèííî¿ àíòðîïîãåííî¿ åì³ñ³¿ ïàðíèêîâèõ ãàç³â çðîñòàòè-
ìå, òî çàãàëüíà ìàñà ïàðíèêîâîãî ãàçó âíàñë³äîê çàçíà÷åíèõ âèùå çâîðîòíèõ çâ�ÿç-
ê³â äîñÿãíå ïåâíî¿ êðèòè÷íî¿ âåëè÷èíè, ùî çóìîâèòü ñàìîïîñèëåííÿ ïàðíèêîâîãî
åôåòó, ÿêèé ñïðè÷èíèòü á³ôóðêàö³éíèé ïåðåõ³ä êë³ìàòè÷íî¿ ñèñòåìè Çåìë³ ó ñòàí
ñàìîðîç³ãð³âó ç íåîáìåæåíèì çðîñòàííÿì ñåðåäíüî¿ òåìïåðàòóðè çåìíî¿ ïîâåðõí³.
Êëþ÷îâ³ ñëîâà: ïàðíèêîâèé ãàç, ïîçèòèâíèé çâîðîòíèé çâ�ÿçîê, ï³äâèùåííÿ òåì-
ïåðàòóðè, á³ôóðêàö³éíèé ïåðåõ³ä.
Introduction. Many systems have a cri-
tical parameter when a system passes from
its original state into a qualitatively new sta-
te with new characteristics. In this paper we
show that at a certain point or threshold when
greenhouse gases reach a critical mass, the
earth�s climate system could change to a qu-
alitatively new state with potentially catast-
rophic consequences for humankind and the
nature. A key driver leading to such a state
is a positive feedbacks upon the earth�s cli-
mate system. There are several studies ad-
dressing the impact of such positive feed-
backs on the greenhouse effect [Lashof et
al., 1997; Torn, Harte, 2006; Scheffer et al.,
2006] but we argue that many researchers
do not give sufficient attention to the strength
and power of this impact. We argue further
that it is such positive feedback that is the
main cause of rapid climate change.
The theory of feedback states the follo-
wing. A feedback works in a system only if
this system is active. It means that the sys-
tem is able to amplify any external action
on it out of its self-energy. This condition
is called the energy criterion [Raisbeck, 1954].
The earth�s climate system satisfies this cri-
terion. Therefore, all conclusions by the the-
DOI: 10.24028/gzh.0203-3100.v40i1.2018.124016
ON THE CRITICAL MASS OF GREENHOUSE GAS
Ãåîôèçè÷åñêèé æóðíàë ¹ 1, Ò. 40, 2018 71
ory of feedback can be used to research
the influence of feedbacks on the earth�s
climate system.
To the possibility of these disastrous con-
sequences have pointed ut earlier [Klimchuk,
Tarasov, 2005]. Now we want to return to
this question making some additions.
Analysis of feedback. An active system
of any kind with feedbacks is shown sche-
matically in Fig.1, where 1 is active system
itself, 2 is the feedback unit, 3 is the sum-
ming unit, U1 is the input action (signal),
U2 is the response of the system to the in-
put signal Uin , Ufu is the response (signal)
of the feedback, and U in is the resulting
action (signal) at the input of the system.
The core of the feedback process is as fol-
lows. A part of the output response U2 of
the system comes through the unit of feed-
back in the form of Ufu to the summing un-
it, where it is added to the input action U1 ,
forming the resulting input signal U in . If the
signs (phases) of signals U1 and Ufu coinci-
de, the signal Uin and the response U2 in-
crease accordingly to feedback. Otherwise
a feedback is negative. The equality of sig-
nals U1 and Ufu means that the internal con-
trol or self-control is realized by the Ufu sig-
nal in systems with feedbacks. This is the
essence of the principle of feedback.
The input action U1 and the response U2
of the system may have different physical
nature. The signals U1 and Ufu are always
similar. The latter is implemented, when ne-
cessary, by the appropriate transformation
of the nature of U2 into a unit of feedback.
Therefore, in considering the earth�s clima-
te system when the input action U1 is un-
derstood as the increase of anthropogenic
emissions of greenhouse gases, we can spe-
ak of positive feedbacks only when there
are such responses Ufu that represent incre-
ases of secondary emissions of greenhouse
gases generated in the earth�s climate sys-
tem due primarily to anthropogenic emissi-
ons of greenhouse gases. These positive fe-
edbacks are discussed below.
It is easy to show that the transmission
factor (function) K fu of the system with a
feedback shown by the structural scheme
in Fig. 1, is expressed by the following ratio
[Bode, 1945]:
,
11
00
0
0
1
2
fb F
K
S
K
K
K
U
U
K =
−
=
γ−
== (1)
where K0 = U2
/
U1 is the transmission factor
(function) of the system without a feedback,
γ = Ufu
/
U2 is the transmission factor (func-
tion) of a unit of feedback, S = γK0 is the lo-
op amplification (reversion ratio) in the fe-
edback loop in open position, and F is the
recurrent difference.
In the ratio (1) it is assumed that the trans-
mission functions of the summing unit 3 on
signals U1 and U fu are equal to unity. The
function γ is otherwise called the feedback
factor and in the case of a positive feedback
is usually denoted by α and in case of a ne-
gative feedback by β.
If the output of the unit of feedback 2 is
disconnected from the adder 3 and the unit
action is used at the input of system 1, the
response at the output of the unit of feed-
back will be numerically equal to the loop
gain S, which in case of multiloop feedbacks
is usually called the reversion ratio. In rese-
arch on the feedback theory the reversion
ratio is usually denoted by T. We back out
of this rule as in what follows we denote tem-
perature by this letter.
In structural scheme in Fig. 1 for the open
feedback loop:
.
11
2
2
0
ufuf
U
U
U
U
U
U
KS ==γ= (2)
Taking into account that in case of a po-
sitive feedback the signs (phases) of signals
U1 and Ufu coincide and for a negative fe-
Fig. 1. Schematic structure of feedback.
E. F. KLIMCHUK, V. F. TARASOV
72 Ãåîôèçè÷åñêèé æóðíàë ¹ 1, Ò. 40, 2018
edback they are different, in the former ca-
se S > 0, in the latter S < 0.
According to the value of S in case of a
positive feedback, 3 modes of operations with
such feedbacks are distinguished.
The first of them, which is called subcri-
tical, is realized when 0 < S < 1, F < 1, Kfu >
> K0 , S = 1, F = 0 correspond to the second
mode called critical. This mode is also called
the critical point or the point of bifurcation.
In this mode the function K fu has infinite
break. Physically it means that the finite re-
sponse U2 of the system is possible at the in-
definitely small input action U1. That is why
its functioning is also possible without U1.
In fact , when S = 1, according to (2), U fu =
= U1, which means that the feedback res-
ponse is an exact copy of the input action.
As a result, the system in a critical mode af-
ter dormancy breaking is able to continue
its movement without the input action U1
because the input action will be replaced by
the response U fu , which is self-sustaining,
being its own cause and effect. Consequen-
tly, when S = 1, qualitative changes occur
in the system. It passes from the mode of ex-
ternal action into the mode of self-trigger-
ing (self-exciting oscillation mode) and its
movement becomes free self-sustaining, that
is , a self-movement. This self-movement is
stationary and takes place at a constant spe-
ed defined by the initial conditions.
And at last the third mode, called super-
critical , is realized when S > 1. Under this
mode the systems with a positive feedback
are in a state of dynamic instability, under
which the differential equations describing
such systems have divergent solutions. As
in every cycle of self-triggering the signal
U fu increases by the factor of S > 1, after
dormancy breaking the self-movement of
such systems exponentially evolves to the
point of their self-destruction or transition
into some stationary state. The latter is pos-
sible only if at some value of the output re-
sponse U2 the partial derivative S′ = ∂S
/
∂U2
becomes negative. As a consequence, as U2
increases further, the reversion ratio will fall
to the value S = 1 and the system will pass
into a state of stationary self-movement with
the limiting value U2 = Uc = const. If for
some reasons U2 exceeds Uc, the reversi-
on ratio will become subcritical (S < 1). Self-
triggering in the system will stop and as a
result of self-braking the system will return
to the stationary state U2 = Uc with S = 1. If,
on the contrary, U2 decreases, the reversion
ratio will become supercritical (S > 1). As a
result of this, self-triggering will increase and
the self-acceleration of the system will re-
turn to the stationary state again with S = 1.
Therefore, at any deflections of U2 from Uc
the system independently comes back to a
stationary state. In equations of motion of
such self-regulation systems these states cor-
respond to limit stable cycles that are called
attractors in mathematics. No matter how
these cycles are reached, their realization
is equivalent to the availability of internal
self-regulation by means of negative feed-
backs in the systems.
Examples of systems with or without at-
tractor cycles are nuclear reactors and ato-
mic bombs. The stationary mode of the for-
mer at the preset power levels is realized by
the systems automatically controlling the cri-
tical value of the reversion ratio S = 1, which
in this case is called the multiplication con-
stant of secondary neutrons. There is no in-
ternal self-regulation in atomic bombs. That
is why in atomic bombs if S > 1, the evolu-
tion of chain self-sustaining fission reactions
tends towards unlimited growth, which le-
ads to their self-destruction through explo-
sion. The mass of the fissile material neces-
sary for the bifurcation transition to the con-
dition of the mentioned chain reaction to oc-
cur is called critical.
Discussion. Returning to the earth�s cli-
mate system, we consider the increase of
GhG emissions to be the input action for the
earth�s climate system, and the increase of
mean temperature of the earth�s surface to
be the response. Then the function of the
transmission of the climate system without
taking into account its feedbacks can be de-
scribed in the following way:
,
0
0
0 V
T
K
∂
∂
= (3)
ON THE CRITICAL MASS OF GREENHOUSE GAS
Ãåîôèçè÷åñêèé æóðíàë ¹ 1, Ò. 40, 2018 73
where ∂V0 is the increase of the primary
GhG emission measured in carbon dioxide
equivalent during the same time interval,
∂T0 is the increase of mean temperature in-
duced by this emission and attributed to du-
ring the same time interval. By the prima-
ry GhG emission we understand the emis-
sion of carbon dioxide from human activi-
ties (anthropogenic emission).
Therefore, the time interval is not inclu-
ded explicitly in expression (3). As CO2 is an
essential part of GhG emissions, it is reaso-
nable to express both primary and secon-
dary GhG emissions in their carbon dioxi-
de equivalents.
In a previous work [Klimchuk, Tarasov,
2005] we singled out 4 main loops of a posi-
tive feedback in the earth�s climate system.
These 4 loops result from secondary GhG
emissions caused by the increase of mean
temperature due to primary GhG emissions.
We think the positive feedback loop, cau-
sed by the secondary emission of water va-
por from open reservoirs, to be the first and
most important for the following reasons.
First, water vapor is the most active green-
house gas, which absorbs all energy of the
infrared spectrum (i. e. the heat) of the earth
and prevents its flowing into space. The
open reservoirs occupy more than 70 % of
the earth�s surface. Second, the primary an-
nual water vapor emission amounts to nearly
600 billion tons, which is 1000 times as much
as the summed annual natural and anthro-
pogenic carbon dioxide emissions. There-
fore, the rise of mean temperature caused
by primary GhG emissions results in the in-
crease of evaporation from open reservoirs,
that is, in the secondary water vapor emis-
sion which in turn increases mean tempera-
ture due to the accumulation of atmospheric
heat energy. In this way an appropriate po-
sitive feedback loop is created, which leads
to the regenerative self-amplification of the
greenhouse effect.
When the water temperature rise the so-
lubility of gases in water falls. This results
in the secondary emission from the world�s
oceans. The amount of CO2 dissolved in the
world oceans is 55 times as much as the
amount of atmospheric CO2. We believe
that the emission of CO2 from the world�s
oceans is the second loop of a feedback in
the earth�s climate system.
However, recent experimental data have
shown that such emissions are not only la-
cking but that the world�s oceans absorbs
about two thirds of anthropogenic CO2 emis-
sions. It is for this reason that biologists ha-
ve raised the alarm about the possibility of
a substantial increase in oceanic acidity. Un-
der present conditions the world�s oceans
are powerful inhibitors of the rate of anth-
ropogenic global warming. However, it will
be shown below that secondary CO2 emis-
sions from the world�s oceans
do exist
but
appear in a very specific way.
We think the positive feedback loop ca-
used by secondary CH4 emissions from the
permafrost zones, where its storage in the
form of gas-hydrates is enormous, to be the
third. The third positive feedback loop is
caused by secondary CH4 emission from the
permafrost zones.
The fourth positive feedback loop results
from the secondary decrease in the earth�s
reflectivity (albedo) due to continuing re-
duction in the area of ice and snow cover.
The decrease in the earth�s reflectivity
leads to the temperature rise of the earth�s
surface. It can be considered as a virtual
emission of an equivalent value of CO2 which
gives the same temperature rise as the al-
bedo decrease does.
We consider the impact of other possib-
le feedbacks on the earth�s climate system
as insignificant.
Before defining feedback factors for the
four positive feedback loops it is necessary
to return to the problem of CO2 absorption
by the world�s oceans. We will consider the
records of ice core air bubbles obtained in
ice coring at the Russian station Vostok in
Antarctica, which generally agree with the
data from a similar coring on the Concor-
dia from the European Project for Ice Cor-
ing in Antarctica (EPIKA) program. In the
article [Petit et al., 1999] provide graphs of
time dependence for the surface air tempe-
rature of atmospheric CO2 and CH4 concent-
E. F. KLIMCHUK, V. F. TARASOV
74 Ãåîôèçè÷åñêèé æóðíàë ¹ 1, Ò. 40, 2018
rations over 420.000 years. These graphs
imply that the earth�s climate is subject to
cyclic changes of up to about 100.000 years
per cycle. We agree with researchers who
consider the relevant variation in the amo-
unt of solar radiation reaching the earth as
the main cause of such cyclicity. Indeed, the
solar cycles contributed to the rise of CO2
concentration. But it does not prove that the
amount of this concentration was defined only
by the intensity of solar energy at that time.
The graphs how that at every increase
in mean temperature, the atmospheric CO2
and CH4 concentrations rise. We are sure
that at the same time the water evaporation
increased and the earth albedo decreased.
Consequently, in the past the earth�s clima-
te was affected by the three ( ignoring any
anthropogenic emissions at that time) men-
tioned positive feedback loops leading to
the regenerative self-amplification of the gre-
enhouse effect. As a result, actual changes
in mean temperature exceeded greatly its
initial variations caused by changes in amo-
unts of solar radiation merely. The main dif-
ference from today was that the primary in-
fluence on the earth�s climate system were
variations in solar radiation, the amplifica-
tion of which led to increases in mean tem-
perature, which turned the world�s oceans
into sources of secondary CO2 emission. That
is to say, when mean temperatures increa-
sed, the value of CO2 equilibrium partial
pressure or its saturation pressure increased,
resulting in CO2 emissions.
Today anthropogenic CO2 emissions are
having a major impact on the climate system
as a result of the partial pressure of CO2 ex-
ceeding the equilibrium value correspond-
ing to the present mean temperature, which
has led to the world�s oceans absorbing ex-
cess of CO2 from the atmosphere. However,
owing to the increase in mean temperatu-
re caused by anthropogenic GhG emissions,
the equilibrium value of CO2 partial pres-
sure corresponding to mean temperature al-
so continuously increases, which is decrea-
sing the oceans� absorption of this gas. In
this way a CO2 positive feedback loop is cre-
ated, with the secondary CO2 emission be-
ing influenced by this decrease in absorp-
tion. Thus, the secondary CO2 emission from
the oceans occur. But they are smaller as so-
me of CO2 is absorbed by the world�s oceans.
That is why by the increase in the prima-
ry GhG emission in (3) its effective value is
understood which is equal to
,car00 VVV ∂−′∂=∂ (4)
where ∂V0′ is the actual increase in the pri-
mary equivalent GhG emission at the fix-
ed time interval, ∂Vcar is the portion of CO2
from the primary GhG emission absorbed by
the World ocean at the same time interval.
Taking into account the above-mention-
ed, the feedback factors for the four consi-
dered positive feedback loops can be pre-
sented as follows:
,
0
car
car T
V
∂
∂
=α ,
0
var
var T
V
∂
∂
=α ,
0
met
met T
V
∂
∂
=α
,
0
al
al T
V
∂
∂
=α (5)
where ∂Vcar, ∂Vvar , ∂Vmet, ∂Val are the se-
condary equivalent emissions caused by the
primary increase in mean temperature ∂T0
and resulting from the secondary emission
CO2, of water vapor, CH4, and the albedo
decrease respectively.
The correctness of the factor αcar is de-
termined by the fact that the absorption of
CO2 by the World Ocean is allowed for by (4).
Based on this and taking into account (2)
and (3) the positive feedback factor αgh and
the reversion ratio Sgh of the greenhouse ef-
fect are defined as:
,almetvarcargh α+α+α+α=α
,
0
gh V
V
S
∂
∂
= Σ (6)
where Σ∂V is the increase of the summed
equivalent secondary GhG emission.
From (6) it follows that the reversion ra-
tio Sgh can be called the factor of seconda-
ry GhG emission.
Based on (1) the transmission function of
the climate system subject to the possible
ON THE CRITICAL MASS OF GREENHOUSE GAS
Ãåîôèçè÷åñêèé æóðíàë ¹ 1, Ò. 40, 2018 75
regenerative self-amplification of the green-
house gas emission becomes:
=
α−
=
∂
∂
=
0
0
0 gh
gh
gh 1 K
K
V
T
K
,
1 ghgh
00
F
K
S
K
=
−
= (7)
where ∂Tgh is the actual increase in mean
temperature subject to the positive feedback
influence caused by the increase ∂V0 in the
primary effective GhG emission, Fgh is the
recurrent difference of the greenhouse ef-
fect.
Taking into account (3), the connection
between the primary increase and the actu-
al one in mean temperature in the presen-
ce of the regenerative self-amplification of
the greenhouse effect is expressed as fol-
lows:
,
gh
gh
0
0
0 FV
T
V
T
∂
∂
=
∂
∂
,
gh
gh
0
F
T
T
∂
=∂
.
gh
gh
0
T
T
F
∂
∂
= (8)
Ñonclusion. Based on the reasoning abo-
ve we come to the following conclusion. The
observed rapid rise of the earth�s mean tem-
perature is caused by two factors. The first
is the increase of the solar activity [Schaller
et al., 2014]. It increases the mean tempera-
ture of the earth. But this temperature rise
leads to the emission of greenhouse gases
from the permafrost zones, increases water
evaporation and decreases the albedo. To
all this is added the greenhouse gas from
man�s productive activity. We consider them
the primary greenhouse gases. Therefore,
the Sun and man�s activity caused the ap-
pearance of primary gases. These gases ac-
count for the second part of the mean tem-
perature rise due to self-triggering of the
earth�s climate system owing to positive fe-
edbacks. This part of the temperature rise
is the second factor of the observed gene-
ral temperature rise.
Indeed, from (8) it follows that if now
Sgh > 0, Fgh < 1, then ∂Tgh > ∂T0. According
to (6), the latter is possible provided the sum-
med secondary GhG emissions are commen-
surable with the effective value of the pri-
mary GhG emissions. The calculations in
[ Torn, Harte, 2006; Scheffer et al., 2006]
suggest such a possibility, making the situ-
ation dangerous. We think that positive fe-
edback gives rise to a danger more serious
than its impact of the earth�s temperature
only.
Increase in secondary GhG emissions ca-
used by the same primary increase in mean
temperature continues to occur as the abso-
lute value of the latter becomes larger. From
the graphs in [Petit et al., 1999] it follows
that over the latest periods of global warm-
ing (in accordance with graphs) the concent-
ration of atmospheric CO2 has not exceed-
ed 0.03 % but now CO2 has 0.038 % and
continues to rise rapidly. Furthermore, the
earth is close to �the zero mark� of a regu-
lar period of global warming resulting from
the earth�s increased absorption of solar po-
wer. Consequently, mean temperature can
only continue to rise in the future. At this
shows of date facts of global warming (IPCC
� Intergovernmental Panel on Climat Chan-
ge). That is why it is inevitable in the com-
ing decades when the summed secondary
GhG emission equalizes the primary GhG
emission, i. e., the factor of the secondary
greenhouse gas emission equals amounts to
the critical value Sgh = 1. The further incre-
ase of this factor will lead to the bifurcation
transition of the earth�s climate system in
which self-triggering of the secondary GhG
emissions results in a state of self-heating
with the tendency for mean temperature to
rise indefinitely. This process can enter a
stationary state only after the complete eva-
poration of the earth�s entire water surface.
In such a case, the pressure on the earth�s
surface will reach 300 atm and the tempe-
rature will exceed 500 °C. The total mass of
atmospheric GhG with which the mentioned
bifurcation will take place we call critical.
The potentialities of the greenhouse effect
are clearly demonstrated by planet Venus.
E. F. KLIMCHUK, V. F. TARASOV
76 Ãåîôèçè÷åñêèé æóðíàë ¹ 1, Ò. 40, 2018
On the critical mass of greenhouse gas
E. F. Klimchuk , V. F. Tarasov, 2018
In recent years, changes in the Earth�s climate have raised concern all around the
globe. Climatologists have been drawing connections between global warming and a
growing number of natural disasters, unexpected temperature fluctuations in some re-
gions of the world and a number of other climatic aberrations. Within the scientific
community, the opinions as to the nature and mechanism of the Earth�s climate chan-
ge have split. Some contend that since the beginning of the industrial revolution, the
carbon dioxide levels in the air have been steadily rising due to human production ac-
tivities. Along with other gases, carbon dioxide has been inculpated for the greenho-
use effect. In connection with this, a number of recent international confeences have
adopted resolutions to reduce carbon dioxide emissions into the atmosphere. Another
group of climatologists bases its findings on observations of solar activity, arguing that
global warming is caused by a recurring spike in solar activity, with the current incre-
ase due to end soon, potentially giving way to a new ice age down the road. In the fol-
lowing work, we put forth yet another hypothesis regarding global warming. The in-
fluence of four main positive feedback loops caused by the secondary emission of wa-
ter vapor, CO2, CO4, and decreased albedo on the earth�s climate system is shown on
the basis of the general theory of feedback. If the present level of primary anthropo-
genic emissions of greenhouse gas (GhG) keeps, the total mass of atmospheric green-
house gas can run up to such a critical value that the mentioned feedbacks, which gi-
ve rise to self-amplification of the greenhouse effect, can cause the bifurcation transiti-
on of the climate system to the state of self-heating tending to the unlimited rise of me-
an temperature of the earth�s surface.
Key words: greenhouse gas, positive feedback, temperature rise, bifurcation transition.
Due to the supercritical carbon-dioxide at-
mosphere of this planet, the surface tempe-
rature of Venus is between 430�470 °C with
the pressure on the planet�s surface at 100 atm.
It is accounted for by the fact that in this
case the available zones of heat and cold
will exhibit their persistence under gradual
temperature rise. As a result, some parts of
the earth experience rain showers (dumping
of accumulated water vapor) while others
experience equalizes hurricanes and torna-
does as a result of thermodynamic disequi-
librium.
Every new emission of a primary (anthro-
pogenic) GhG causes the growth (or con-
centration) of the GhG shielding layer which
will reduce the amount of the earth�s heat
radiation escaping into space. This reducti-
on will occur exponentially with the emissi-
on of every new primary GhG portion as the
intensity of any radiation going through the
shielding layer falls exponentially, that is,
the earth�s capacity to shield heat improves.
It means that value K0 = ∂T0 /∂V0, in (3) and
(7) is the increasing time function, which
increases the danger of the early occurren-
ce of this state.
The conclusion to be drawn from this dis-
cussion is the view of some climatologists that
the future temperature rise in the earth�s
atmosphere poses no risk to be dangerous.
It causes our anxiety.
Taking into account inestigations about
impact at climate positive feedback are con-
tining [Plattner et al., 2009; MacDougall et
al., 2015; MacDougall, Knutti, 2016; Rug-
enstein et al., 2016] we propose that rese-
archers calculate estimates of the time de-
pendence ∂Fgh /∂t i. e., the rate of approach
to the self-heating bifurcation and estima-
te the likely time necessary for the transiti-
on to this state under the condition of futu-
re growth of the increase in the primary GhG
emission.
ON THE CRITICAL MASS OF GREENHOUSE GAS
Ãåîôèçè÷åñêèé æóðíàë ¹ 1, Ò. 40, 2018 77
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