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Towards a Physically Motivated Planetary Accounting Framework
M. Barbosa, O. Bertolami, F. Francisco
Departamento de Física e Astronomia and Centro de Física do Porto, Faculdade de Ciências, Universidade do Porto, Rua do
Campo Alegre 687, 4169-007 Porto, Portugal
Abstract
In this work we present, based on the Landau-Ginzburg model of phase transitions, a physically motivated
planetary Accounting Framework for the Earth System. We show that, up to interacting terms among the
impact of the human activity in terms of the Planetary Boundaries variables, our physical formulation is
up to physical constants that relate concentration and flux of substances of the Planetary Boundaries vari-
ables, as proposed by the accounting framework of Kate and Newman with the underlying thermodynamical
transformation quantifiable by the Landau-Ginzburg inspired model.
Keywords: Anthropocene, Earth System, Accounting framework
1. Introduction
The impact of human activities on the Earth Sys-
tem (ES) has become a defining issue of our time and
has given origin to the proposal of a new geological
epoch, the Anthropocene. Given the extent of human
action, measuring its impact in an objective way is
crucial to ensure that fair and rational stewardship
measures are implemented in the near future. The
mounting evidence, not only of anthropogenic climate
change, but also of the deterioration of several ecosys-
tems makes it urgent. Ideally, consensual global ac-
tions along the lines of the 1987 Montreal Protocol to
halt the destruction of the ozone layer and the 1997
Kyoto Protocol for reducing the greenhouse-gas emis-
sions, renewed in Paris in 2015, would be agreed, al-
lowing for the design of strategies for a sustainable fu-
ture with quality of life all humankind. A fair, reliable
and uncontroversial accounting system for human im-
pacts is an indispensable tool for any such effort.
Accounting for the impact of the human action on
the ES has been the purpose of several proposals
such as the I=P AT measure [1], the Kaya iden-
tity [2] and the well developed Ecological Footprint
proposed in the 1990’s [3] which since 2003 is being
carried out in a systematic way by the Global Foot-
Email addresses: up201305930@fc.up.pt (M. Barbosa),
orfeu.bertolami@fc.up.pt (O. Bertolami),
frederico.francisco@fc.up.pt (F. Francisco)
print Network. Although these proposals have sev-
eral virtues, as centred on the socio-economical nature
of human activities, they lack an unequivocal corre-
spondence to the natural biogeochemical and physi-
cal processes of the ES that are now being disrupted.
For instance, the quite accomplished Ecological Foot-
print is based on the confrontation of a measure of
bio-capacity for a given territory, measured in global
hectares per capita, against the overall consumption
of energy, biomass, building material, water, etc, mea-
sured in the same units. Of course, this involves a
quite complex conversion methodology, whose unde-
niable usefulness contrasts with its somewhat arbi-
trary nature, which in turn makes the connection with
the impact on the ES somewhat indirect and opaque.
In order to achieve a more direct method to access
the impact of the human action on the ES, one should
start from the very basis of its functioning principles,
as proposed in Ref. [4].
Recently, we have proposed that the ES transition
from the Holocene to the Anthropocene could be re-
garded as a phase transition and described by the
Landau-Ginzburg theory [5]. The relevant thermody-
namic variable of this approach is the free energy, F,
and it was suggested that the relevant order parame-
ter, ψ, of the description is the relative temperature
deviation from the Holocene average temperature TH,
that is ψ= (T−TH)/TH.
This framework allows for establishing the state
Preprint submitted to July 24, 2019
of the ES in terms of the relevant physical vari-
ables, (η, H ), where ηcorresponds to the astronomi-
cal, geophysical and internal dynamical effects while
Hstands for the human activities.
The proposed physical model allows for obtaining
the so-called Anthropocene equation, i.e., the evolu-
tion equation of the ES once it is dominated by the
human activities, and to show that the transition from
the Holocene conditions to the Anthropocene arises
from the great acceleration of the human activities
that was witnessed by the second half of the 20th
century [6].
It has also been shown how the human activities
function, H, can be decomposed into its multiple com-
ponents, with a straightforward correspondence to the
measurement of human impacts in terms of the Plan-
etary Boundaries [7].
This logic has recently been used to set up an ac-
counting framework that manages to gauge the hu-
man impact in terms of 10 quotas based on the 9
planetary boundary parameters [8]. This procedure
allows for an empirical basis for the environmental
issues and a set of accessible indicators that can be
adopted at various scales by various agents, being thus
a poly-scalar approach.
Subsequently, we performed a phase space analy-
sis of temperature field, (ψ, ˙
ψ)and showed that the
recently discussed Hothouse Earth scenario, [9], cor-
responds to a stable minimum and, therefore, to an
attractor of the trajectories of the dynamical system
that describes the ES [10].
The purpose of the present work is to provide a
physical support to the accounting framework of Ref.
[8]. This paper has the following structure: in the
next section we review our Landau-Ginzburg model
proposal and discuss its main implications for the
description of the ES. In section 3, we discuss how
after splitting the human activity on its impact on
the parameters of the planetary boundaries and disre-
garding interaction terms, our physical framework can
naturally give origin to an accounting framework that
resembles up to multiplicative constants the frame-
work proposed in Ref. [8]. In fact, it is possible to
show, given the generality of our approach, that it
contains, for instance, under certain conditions, the
I=P AT proposal. However, a distinct feature of our
approach is that it allows for considering the interac-
tion terms between the various planetary boundaries,
implying that any accounting framework is useful pro-
vided these interacting terms are neither important
nor evolving in a time scale such that the effect of
these terms can be relevant. In section 4 we con-
sider the interaction between two planetary bound-
aries, namely the concentration of atmospheric car-
bon dioxide and the ocean acidity. In section 5 we
present our results. Finally, in section 6 we discusss
our conclusions.
2. The Physical Model
The fundamental insight of our physical model of
the ES is its description of the transition from the
Holocene to the Anthropocene as a phase-transition
through the Landau-Ginzburg Theory [5]. The main
thermodynamic variable is the free energy,
F(η, H ) = F0+a(η)ψ2+b(η)ψ4−h(η)Hψ, (1)
where coefficients a(η),b(η)and h(η)depend on the
set of variables ηand the effect of human intervention,
H, is introduced as an external field modelled as linear
term in the order parameter. The effect of this term
is a clearly destabilizing one, as it is easy to show that
the model predicts that, in the Anthropocene,
hψi ≈ H
4b1/3
,(2)
which means the equilibrium temperature of the ES,
represented by hψi, grows proportionally to the cubic
root of human activities [5].
It has also been shown that the dynamical model
arising from this description has an attractor of tra-
jectories, provided that human action, H, is finite and
some damping is introduced into the ES model. This
critical point of the dynamical system corresponds
necessarily to an ES trajectory towards a minimum
where the temperature is greater than during the
Holocene equilibrium [10].
It is relevant to point out that these results are
consistent with the features of the ES trajectories in
the Anthropocene arising from the qualitative discus-
sion of Ref. [9]. Clearly, this means an increase in the
global temperature which can lead to a chain failure
of the main regulatory ecosystems of the ES, that al-
ready show tipping point features [9].
Given that the purpose of an accounting framework
is to gauge the impact of the human drivers, H, with
respect to the Holocene conditions expressed by the
first three terms in the free energy, we can isolate the
2
term describing human activities as the change in the
ES free energy in the Anthropocene,
∆F(H)|Anthro =−Hψ, (3)
where we have absorbed the constant, h, into the defi-
nition of H. This can be safely done because the time
scale of human impacts is so much faster than that of
natural driven change to the ES.
The effect of human activities in altering the op-
timum Holocene conditions can be decomposed into
its different components, denoted by hi. These com-
ponents have a straightforward correspondence to the
set of 9 or 10 parameters that compose the Planetary
Boundaries (PB) framework [7]. In the context of
the PB, these parameters specify the state of the ES
in terms of its deviation from the Holocene-like con-
ditions that sustain human civilisation as we know
it. The maintenance of these conditions is ensured if
the ES remains within the so-called Safe Operating
Space (SOS), which sets a limit on each of the PB
components [11]. The function Hdescribing human
intervention can thus be written as [10]:
H=
9
X
i=1
hi+
9
X
i,j=1
gijhihj+
9
X
i,j,k=1
αijk hihjhk+. . . ,
(4)
where the second and third set of terms indicate the
interaction among the various effects of the human
action on the PB parameters. Of course, higher or-
der interactions terms can be considered, but we shall
restrict our considerations up to second order and,
in fact, to a subset of planetary boundary param-
eters. It is physically reasonable and mathemati-
cally convenient to assume that the 9×9matrix,
[gij]is symmetric, gij =gj i, and non-degenerate,
det[gij]6= 0. In principle, these interactions terms are
sub-dominating, however, their importance has to be
established empirically. As discussed in Ref.[10], they
can lead to new equilibria and suggest some mitiga-
tion strategies depending on the sign of the matrix
entries, gij, and their strength [10].
Indeed, in order to understand these possibilities,
consider only a couple of parameters, say h1and h9,
and assume, in particular, that the ninth parameter
corresponds to the Technosphere [12], i.e., the set of
human technological activities aiming to repair or to
mitigate the action on the variables away from the
SOS , as suggested in Ref. [10]. Hence, we can write
Has,
H=h1+h9+ 2g19h1h9+g11 h2
1+g99h2
9.(5)
It is easy to see that if h1>0,h9<0and g19 >0,
then the effect is to mitigate the destabilizing effect of
h1. The net effect of this technological interaction is
to ensure that the minimum due to human interven-
tion is, as discussed in Ref. [5], closer to the Holocene,
minimizing the departure of the ES temperature from
the one at the Holocene. We could also argue that g11
and g99 might be negative too, given that they can
have a saturating effect on themselves.
3. The Accounting Framework
There is a natural accountancy criterion built into
the physical framework discussed in the previous sec-
tion, given that it allows for the quantification of the
destabilizing effect of human intervention on the ES.
This can be exemplified when we consider the deple-
tion of living biomass stock, as discussed in Ref. [5].
Considering this as the sole component of human in-
tervention,
H=α∆mLB,(6)
where mLB is the total living biomass and α= 3.5×
107J/kg is the conversion constant of biomass into
energy. Estimates indicate that the amount of living
biomass has dropped from 750 ×1012 kg in 1800, to
660 ×1012 kg in 1900, and to 550 ×1012 kg in 2000
[13, 14].
This biomass depletion is well correlated with, e.g.
CO2concentration, hence it suggests a quota, QBM ,
for a country or region at a fixed time interval, ∆t,
by multiplication by a fraction, f(A,P,GDP, . . .), of
the global quota that is a function of the territory’s
area, population, GDP or other relevant measures,
QLB =α∆mLB
∆tf(A,P,GDP, . . .),(7)
Of course, the specific planetary boundary breakdown
suggests itself a set of quotas:
Qhi=hi
∆tf(A,P,GDP, . . .),(8)
which requires an accurate knowledge of all hi‘s and
their scaling down to a country (region). Further-
more, the usefulness of this approach is valid as far
as the contribution of the interaction terms is sub-
dominant at the time interval ∆t, otherwise, this set
of quotas is meaningless in this time interval.
The temperature field can have, as discussed in
Ref. [5], a spatial dependence and thus the free en-
ergy should have terms like |∇ψ|2and ∇2ψ. These
3
terms could be split in contributions from countries
(regions) area and hence should be normalized by
their area. This suggests the following quota system-
atics for a given country (region):
QSpatial =|∇ψ|2
∆tf(A),(9)
or an equivalent expression for the Laplacian.
Hence it is clear that our physical framework sug-
gests several strategies for an accountancy procedure.
Neglecting interactions, the breakdown of H into its
planetary boundary components leads, for a partic-
ular choice of the fraction, f(A, P, GDP, . . .)to the
Planetary Boundary Accounting Framework of Ref.
[8] up to a multiplicative constant that converts the
matter concentration and flows of each of the plane-
tary boundary quantities into the free energy involved
in the respective set of thermodynamic processes. For
sure, these processes must be understood and decom-
posed in its most elementary steps so that their very
essence is captured and quantified.
In fact, it is fairly easy to show that our proce-
dure encompasses other accounting proposals through
a specific choice of the subset of terms of the break-
down of H in its Planetary Boundary (PB) compo-
nents. For instance, if we consider the Technosphere,
as discussed above, it is reasonable to assume that
its impact must be conjugated with the population,
P, and affluence, A, or GDP, which implies that a
particular term of our breakdown of H contains the
I=P AT proposal. Of course, since the I=P AT
measure is purely socio-economical, for instance, Tis
measured in the number of patents, it is somewhat
disjoint from the thermodynamical nature of our ap-
proach, but there is a clear parallelism of the two
measures once a suitable planetary boundary is cho-
sen.
It is relevant to point out that none of the account-
ing systems proposed so far includes the interactions
terms and some among these might be clearly impor-
tant. If so, the next issue is to consider the typical
time scale for a given planetary boundary parameter
to affect the others. For instance, in what concerns
the relationship between the CO2concentration and
its effect on the acidity of the oceans a typical time
scale of change is about 240 days [15]. The under-
standing of the underlying processes associated to this
change will be discussed in the following section. If
this time scale is the smallest one involving the inter-
action between the planetary boundary parameters,
then the accounting system of Ref. [8] yields a reli-
able picture of the ES at this time scale.
4. Modelling the interaction between Plane-
tary Boundaries
As discussed in the previous section, for each PB,
there is a control variable that measures its deviation
from the Holocene conditions. Each of these control
variables, xican then be associated to an hiterm
in the human activities function given by Eq. (4) by
means of a proportionality constant αi. The corre-
spondence is thus given by,
hi=αi∆xi.(10)
The αiproportionality constants reflect the physical
processes of each PB and have units such that the hi
terms have dimensions of energy.
We present here the modelling of the interaction be-
tween climate change and the ocean acidification. We
will denote these with indexes i= 1,2, respectively,
and the interaction term between them by g12.
In both these cases, the control variables, x1, x2,
have units of concentration, namely, Carbon Dioxide
concentration in the atmosphere, for climate change,
and H+ion concentration in seawater, for ocean acid-
ification.
These choices allows for writing the PB in the fol-
lowing way:
h1=α1∆x1=α1(x1−x1,H+),(11)
h2=α2∆x2=α2(x2−x2,H),(12)
where is the direct anthropogenic carbon emission in
a given time period and xi,Hcorrespond to Holocene
values.
To calculate the interaction term g12 we examine
two different scenarios, the non-interacting and the
interacting ones. In the first scenario, since there is
no interaction there is no change in h2due to carbon
dioxide concentration increase, so the global variation
is equal to the change in h1.
∆H=α1(x1−x1,H+) + α2(x2−x2,H).(13)
On the other hand, if we consider g12 6= 0, it is
known that the variation of Hover the time scale
long enough to achieve a quasi-static equilibrium is
∆H=α1(x1−x1,H) + α2(x2−x2,H)+
+α1δx1+α2δx2,(14)
4
where δx1and δx2are changes in the concentra-
tions of carbon and hydrogen ion respectively, due to
the existence of interaction. We shall compute these
quantities later. All factors considered, the propo-
sition is that if we add the interaction term to the
equation that resulted from the non-interaction sce-
nario we should get the equation from the interaction
scenario, yielding:
∆H(g12 = 0) + 2g12 h1h2= ∆H(g12 6= 0),(15)
which yields an equation for the value of the interac-
tion term,
+ 2g12h1h2=α1δx1+α2δx2,(16)
that we can solve to obtain an expression for the in-
teraction term, g12, from empirical data,
g12 =α1(δx1−) + α2δx2
2α1α2(x1−x1,H+)(x2−x2,H).(17)
The previous equation shows that the existence of an
interaction between the carbon dioxide in the atmo-
sphere and the ocean acidity leads to perturbations,
δxi’s, of the control variable. But we still need to
develop a method that yields these variations as an
output given the annual carbon emission values as an
input. It is here that the knowledge of the carbonate
system is particularly important.
First, the relation between the concentration of
CO2in the atmosphere and the oceans is regulated
by a chemical equilibrium, with a solubility constant
K0,
CO2(g.)CO2(aq.).(18)
Then, in the ocean, there is a well known chain of
chemical reactions that describes the carbonate sys-
tem and may be broken down into two simple reac-
tions,
CO2+ H2Ok1
¯
k1
HCO−
3+ H+,(19)
HCO−
3
k2
¯
k2
CO2−
3+ H+,(20)
where the kiare the forward reaction rate and ¯
kithe
reverse reaction rate coefficients. We now discuss the
dynamics involved in the carbonate system and how
the increase of carbon dioxide in the atmosphere due
to anthropogenic carbon emissions leads to the acidi-
fication of the ocean.
First we need to describe mathematically how the
concentrations of all inorganic forms in the ocean vary
in time after the system’s equilibrium is disturbed,
that is, when there is a gradual injection of carbon
dioxide into the system. Recalling Eqs. (19) and (20),
and introducing the following notation, v= [CO2],
w= [H+],y= [HCO−
3],z= [CO2−
3], then:
dv
dt =−k1v+¯
k1yw, (21)
dy
dt =k1v−¯
k1yw −k2y+¯
k2zw, (22)
dz
dt =k2y−¯
k2zw, (23)
dw
dt =k1v−¯
k1yw +k2y−¯
k2zw. (24)
Note that, above, we defined x1as the CO2concen-
tration in the atmosphere and assigned it as the con-
trol variable for climate change. Now, we are using v
as the CO2concentration in the oceans. Since these
two concentrations follow an equilibrium dictated by
a solubility constant, they can be interchanged as the
control variable for climate change.
We point out that Eq. (21) is the time evolution
of carbon dioxide’s concentration simply due to the
chemical dynamics, but it is of our interest to study
how the entire system reacts if there is a continuous
input of carbon dioxide, even though by small quan-
tities at a time. From Ref. [16] we obtain that over
the last year (2018) the average concentration of car-
bon dioxide in the atmosphere has been increasing
almost linearly1, with a rate of 0.250 ppm per month
(from 407 ppm in January 2018 to 410 ppm in Jan-
uary 2019).
In Ref. [15], it is estimated that the time it takes
for the exchange between gaseous and aqueous car-
bon dioxide to reach an equilibrium is about a year
(240 days). With that in mind, we consider that at
the start of an year (e.g January 2018) carbon dioxide
exchange has reached an equilibrium and an atmo-
spheric concentration of currently 407 ppm, and con-
sider a steady input of carbon such that at the start
of the following year (in this scenario January 2019)
the increase in concentration was of 3ppm granting a
total of 410 ppm, meaning that there was an increase
of about 0.737% of CO2in the atmosphere. Over the
year the oceans must have reached a new equilibrium
1This linear approximation might be a crude one, but it is
still interesting to see the outcome of this simple scenario.
5
and since the percentual increase was small, the new
equilibrium should not be too far away from the old
one. Taking this into account, it seems reasonable to
consider a Linear Stability approach to compute how
the concentrations of the inorganic forms respond to
a steady, but small increase of CO2in the system.
Defining a vector ~r as:
~r =
v
y
z
w
,(25)
and a small perturbation around a fixed ~r,~r →
~r +δ~r. From the linear stability approach ˙
δri=
∂˙ri
∂rj|~r0δrj+O(δr2), where the dot stands for the usual
time derivative and ~r0is the fixed point for which
δ~r = 0. We may represent the derivative terms as
components of a matrix Msuch that Mij =∂˙ri
∂rj|~r0.
The equilibrium condition gives the following concen-
trations:
~r0=
v0
y0
z0
w0
=
v0
k1
¯
k1
v0
w0
k1k2
¯
k1¯
k2
v0
w02
w0
(26)
Given these values the matrix takes the form:
M=
−k1¯
k1w00k1v0
w0
k1−¯
k1w0−k2¯
k2w0−k1v0
w0+k1k2
¯
k2
v0
w02
0k2−¯
k2w0−k1k2
¯
k2
v0
w02
k1−¯
k1w0+k2−¯
k2w0−k1v0
w0−k1k2
¯
k2
v0
w02
(27)
That is written in terms of v0and w0which can
be calculated by knowing the concentration of car-
bon dioxide (v0) and seawater pH (w0) at a given
time. The dynamical equations around the equilib-
rium point are then given by:
˙
δ~r =Mδ~r +mˆex,(28)
where the term mˆexis the external contribution from
the steady increase in the CO2concentration.
5. Results
Even though we have already considered an ap-
proximation for the dynamics, it is still hard to cal-
culate the expressions for the equilibrium concentra-
tions, therefore in this section we shall consider some
numerical values and interpret the results in terms of
these particular values.
First let us take into account the amount of carbon
dioxide that was emitted during 2017 and 2018. It
is estimated in Ref. [17], that in 2017 approximately
36.1×1012 kg were emitted, and in 2018 approxi-
mately 37.1×1012 kg. It is possible to convert these
quantities into concentrations in the atmosphere. The
conversion rate is 1 ppm(CO2)=2.12 ×1012 kg. We
should note that in these two consecutive years there
was an increase of approximately 2.5% in carbon emis-
sion, meaning that, if this trend continues, by the end
of 2019 the total amount of carbon emitted will be
close to 38.0×1012 kg. It is helpful to summarize
some relevant quantities that can be derived from the
previous values.
Table 1: Carbon dioxide annual emission and related values.
Starred values (∗) are predictions based on the extrapolation
of carbon emissions.
Year 2017 2018 2019
CO2emissions (1012 kg) 36.2 37.1 38.0*
Atmospheric CO2
equilibrium
concentration, x1
(ppm)
406 407 410
Oceans CO2
equilibrium
concentration Jan. 1st,
v0(10−5mol/dm3)
1.218 1.221 1.230
Atmospheric CO2
concentration increase,
(ppm)
17.1 17.5 17.9*
Oceans CO2
concentration increase,
m(10−14 mol/(dm3s)
1.62 1.66 1.70*
The molar concentrations in Table 1 can be calcu-
lated using Dalton’s Law for partial pressure and the
value for the solubility constant, K0, was estimated
using the information provided in Ref. [15] assuming
a temperature of 298K, which gives a value for the
solubility constant K0≈0.03 mol/(dm3atm).
Let us now verify whether our model is consistent
and if we can forecast some values for the beginning
of 2020. The average seawater pH in 2017 may have
been very close to 8.07, [15], which indicates an hy-
drogen ion concentration of 8.44×10−9mol/dm3. Re-
calling the matrix Mand the concentrations’ vector,
~r, described in the previous section, it is clear that
in order to calculate the variations of all components’
concentrations we need the numerical values for the
6
equilibrium concentrations of [CO2],v0, and [H+],w0,
as well as the forward and reverse reaction rate con-
stants. The equilibrium concentrations v0are given
in Table 1 and w0is the value calculated from the
assumed pH, namely, 8.07. The change per unit of
time, m, is also shown in Table 1. The reaction rates
are provided in Ref. [18] and are the following:
k1= 3.71 ×10−2s−1,(29)
¯
k1= 2.67 ×104dm3/(mol s),(30)
k2= 59.44 s−1,(31)
¯
k2= 5.0×1010 dm3/(mol s).(32)
According to the definition of the PB’s variables,
we may identify, up to a constant, vwith the at-
mospheric carbon concentration, x1, and wwith the
hydrogen ion concentration in the ocean x2. Now
we substitute the values into Eq. (28) and consider
a time scale of one year, substituting the time, t, by
3.16×107s. Starting from 2017 with v0= 1.22×10−5,
w0= 8.44 ×10−9and m= 1.62 ×10−14, the values
for the variations in the concentrations vand ware:
δv(2017 →2018) = 3.24 ×10−8,(33)
δw(2017 →2018) = 1.84 ×10−11 .(34)
According to the proposed model the concentration
xat the beginning of an year should be the concen-
tration at the start of the previous year v0(assum-
ing that equilibrium was achieved at the time) plus
the small variation δv. So, if in 2017 we start with
v0= 1.218 ×10−5, then in 2018 we should have:
v0(2018) = v0(2017) + δv = 1.221 ×10−5.(35)
From Table1 we see that in 2018 v0was 1.22×10−5
which is fairly close to the value given in Eq. (35).
Also, we get a new value for w0by adding δw:
w0(2018) = w0(2017) + δw = 8.46 ×10−9,(36)
which gives a new value for the acidity level, pH =
8.073. A decrease of 0.0124%.
Now let us do the same for the time period between
2018 and 2019, v0= 1.221 ×10−5,w0= 8.461 ×10−9
and m= 1.66 ×10−14:
δv(2018 →2019) = 3.33 ×10−8,(37)
δw(2018 →2019) = 1.89 ×10−11 .(38)
From these values we can perform the same kind of
computation as before to obtain:
v0(2019) = v0(2018) + δv = 1.224 ×10−5,(39)
which once again is very close to the value given in
Table 1, v0= 1.230 ×10−5up to a 0.49% increase.
The new value of w0is:
w0(2019) = w0(2018) + δw = 8.48 ×10−9,(40)
which gives a new value for the acidity level, pH =
8.072, a decrease of 0.0124%.
So if we consider the values shown in Table 1 for
the carbon emissions during 2019, and recalling that
those values are assuming the same increasing trend
in the amount of emissions, it is possible to forecast
values for the following year 2020. Then, for the time
period between 2019 and 2020: v0= 1.23 ×10−5,
w0= 8.48 ×10−9,m= 1.70 ×10−14:
δv(2019 →2020) = 3.42 ×10−8,(41)
δw(2019 →2020) = 1.93 ×10−11 .(42)
Leading to the following equilibrium values for 2020:
v0(2020) = v0(2019) + δv = 1.233 ×10−5,(43)
assuming an 0.49% error in the previous calculation.
This value of v0means that by the end of the year
(2019) and the beginning of 2020 the carbon dioxide
concentration in the atmosphere would be approxi-
mately (411 ±2) ppm. For the hydrogen ion we get:
w0(2020) = w0(2019) + δw = 8.50 ×10−9,(44)
which gives a new value for the acidity level, pH =
8.07, corresponding to a decrease of 0.0124%. The
same calculations were performed for each year of the
last decade and the results are summarized in Table 2.
To determine the interaction term, g12, we return to
Eq. (17), for which we now have all the values, except
for α1and α2. Since α1,α2and g12 should reflect the
complex dynamics of the ES processes, they would,
at least theoretically, be constant. Since, we are only
looking at the interaction between two processes and
are describing them perturbatively, we should expect
to be able to find a reasonable approximation for the
relationships between these three parameters from the
data in Table 2.
Recalling Eq. (17 and using the data from Ta-
ble 2, we can write multiple equations for g12 by as-
suming that g12 is constant and taking any pair of
7
Table 2: Carbon dioxide annual emission and related values
Year 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
CO2emissions (1012 kg) 33.1 34.4 35.0 35.3 35.6 35.5 35.7 36.2 37.1 38.0* 39.0*
Atmospheric CO2
concentration increase,
(ppm)
15.6 16.2 16.5 16.7 16.8 16.8 16.8 17.1 17.5 17.9* 18.38*
Atmospheric CO2
equilibrium concentration,
x1(ppm)
388 391 393 395 397 399 402 406 407 410 413*
Oceans CO2equilibrium
concentration, v0
(10−5mol/dm3)
1.164 1.173 1.179 1.185 1.191 1.197 1.206 1.218 1.221 1.230 1.233*
Oceans CO2concentration
increase, m
(10−14 mol/(dm3s)
1.48 1.54 1.57 1.58 1.59 1.59 1.60 1.62 1.66 1.70* 1.75*
Oceans H+equilibrium
concentration, w0
(10−9mol/dm3)
8.32 8.34 8.35 8.37 8.39 8.41 8.43 8.44 8.46 8.48 8.50*
δv (10−8mol dm−3)2.89 3.02 3.08 3.12 3.15 3.15 3.18 3.24 3.33 3.42* 3.53*
δw (10−11 mol dm−3)1.69 1.75 1.79 1.80 1.82 1.81 1.82 1.84 1.89 1.93* 1.99*
New oceans CO2
concentration, v0+δv
(10−5mol/dm3)
1.167 1.176 1.182 1.188 1.194 1.200 1.210 1.221 1.224 1.233* 1.240*
New oceans H+
concentration, w0+δw
(10−9mol/dm3)
8.34 8.35 8.37 8.39 8.41 8.43 8.44 8.46 8.48 8.50* 8.52*
New atmosphere CO2
concentration, x1+δx1
(ppm)
389 392 394 396 398 400 403 407 408 411* 415*
New oceans pH 8.079 8.078 8.077 8.076 8.075 8.074 8.074 8.073 8.072 8.071* 8.071*
Atmosphere CO2
concentration error
0.52% 0.25% 0.25% 0.25% 0.25% 0.50% 0.66% ~0 0.49% - -
8
20 000 40 000 60 000 80 000 100 000
α2
α1
0
5
10
15
20
Figure 1: Histogram of the sample of values for the α2/α1ratio
from the data resulting from the years 2010 to 2018. The mean
value is hα2/α1i= 57056.
years and equating their correspondent equation, e.g.
g12(2010) = g12 (2011). Doing this equality for all
available combinations yields a sampling of experi-
mental values for the proportionality constant α2/α1,
the results of which are depicted in Fig. 1.
With this relationship, we can then write g12 as a
function of only one of them. At first glance, it seems
more obvious to write it as a function of α1,
g12 =A
α1
,(45)
because α1is the constant related to the greenhouse
mechanism in climate change, as defined in Eq.(11).
To each value of α2/α1corresponds a values of A,
which leads to the distribution depicted in Figure 2,
with a mean value of −1548.7 mol−1dm3and a stan-
dard deviation of 307.1 mol−1dm3. The fact that the
individual results seem to converge to a single value
is reassuring that the assumptions made so far are
reasonable.
All that remains now is to determine α1. A full
determination would require a knowledge of the a(η)
and b(η)functions that control natural drivers of the
ES in Eq. (1). Still, we can use the biomass account-
ing described in Section 3 as a rough estimate for the
purposes of getting orders of magnitude for the differ-
ent terms. Thus, we take the average annual rate of
biomass depletion at 1013 kg/year, of which around
1012 kg/year is living biomass, and its energy con-
tent 3.5×107J/kg. The conversion of living biomass
into carbon dioxide emissions is not straightforward
and depends on its exact composition, but if we as-
sume that carbon atoms make up for the majority of
the mass of organic matter, by comparing the molar
mass of carbon and carbon dioxide, we can estimate
-2500 -2000 -1500 -1000
A0
5
10
15
20
25
Figure 2: Histogram for the Aconstant that determines the
interaction term g12 in terms of α1, as per Eq. (45)
that there will be 3.6 kg if CO2emitted for each kg
of biomass consumed, owing to the incorporation of
oxygen in combustion, that matches the values in Ta-
ble 1.
We can use this to obtain a figure of merit for
α1∼4×1026 J/(mol dm3)that relates the increase in
carbon dioxide concentration in the atmosphere with
energy degradation. We can then use the mean value
for the relationship between α1and α2obtained as
described above. This means that α2∼6×104α1=
2.4×1031 J/(mol dm3)and g12 ∼3.4×10−24 J−1.
These orders of magnitude, by themselves, have little
meaning, since they are comparing different things.
However, we can compare the relative importance of
each hicontribution and the interaction term, g12 .
Thus,
h1=α1∆x1∼4×1026 ×5×10−6=
= 2 ×1021 J(46)
h2=α2∆x2∼2×1031 ×2×10−9=
= 4 ×1022 J(47)
g12h1h1∼ −3.4×10−24 ×2×1021 ×4×1022 =
= 2.7×1020 J(48)
The main takeout from these results is that the
interaction term is at least one order of magnitude
below the other two terms, which mean that it is rel-
evant but should not affect the overall consistency of
an accounting system within the time scale of a year
or so.
6. Conclusions
In this work we have shown that the physical de-
scription of the ES in terms of the theory of phase
9
transitions of Landau-Ginzburg discussed in Refs.
[5, 10] provides a natural accounting framework for
measuring the impact of the human drivers once these
are broken in terms of planetary boundary compo-
nents.
The arising framework suggests a few accounting
strategies, which can be gauged in terms of the pop-
ulation, the area, or the GDP of a given country or
region. We have shown how a quota system can be
built from the particular example of the depletion of
living biomass. A similar systematics would lead to a
quota system based on the planetary boundaries and
it is argued that it closely resembles the quota system
of Ref. [8].
Furthermore, we have discussed the role of the in-
teraction terms between the planetary boundary pa-
rameters and how they render a quota system valid in
a time scale typically closer than the slowest interact-
ing process. We worked out a specific example involv-
ing the interaction of the CO2concentration and the
ocean acidity establishing the procedure to obtain a
description of human action that can have interaction
terms. This theoretical exercise is an important first
step in achieving a useful modelling procedure for the
ES components that can be then inserted into a dy-
namical description of the ES that can establish the
conditions under which it can remain within the Safe
Operating Space.
Acknowledgements
The authors would like to thank Kate Meyer, Will
Steffen and Alessandro Galli for the insightful dis-
cussions. The work of FF is supported by the Fun-
dação para a Ciência e Tecnologia through grant
SFRH/BPD/118649/2016.
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