(William Happer, W.A. Wijngaarden) USofA, Canada - Physics Flow Rates For CO2

Physics Rate Equations W. Happer and W. A. van Wijngaarden June 16, 2020

1

Introduction

Ed Berry has written a clear and interesting paper, The Core Issues of the Human Carbon Cycle[1]. Its basic assumptions are the Physics Rate Equations, the four coupled differential equations, (23) – (26). We can summarize those four equations as a single matrix equation, d |L⟩ = −Γ|L⟩ + |H⟩, dt

(1)

Ed has assumed that Earth’s exchangeable carbon includes an amount L1 = Lg in the land, an amount L2 = La in the atmosphere, an amount L3 = Ls in the shallow ocean, and an amount in L4 = Ld in the deep ocean. These amounts can be thought of as the elements of a column vector,     L1 Lg  L2   La     |L⟩ =  (2)  L3  =  Ls  .

L4 Ld The values of Lk can be given in any unit that is proportional to the number of carbon atoms, for example, per cent of total exchangeable carbon (%), or petagrams of carbon (PgC). The indices k = 1, 2, 3, 4 of Lk will be called reservoir indices, since they designate one of the four carbon reservoirs of the model.

The last term of (1) describes human addition rate of carbon to the reservoirs,     H1 0  H2   Hfa     (3) |H⟩ =   H3  =  0  .

H4 0 As shown by the column vector on the right of (3) Ed has assumed no direct addition of anthropogenic carbon to the land, the shallow ocean or the deep ocean (H1 = H3 = H4 = 0).

He assuned that combustion of fossil fuels adds new exchangeable carbon to the atmosphere at a rate Hfa. Ed’s physics rate equations also included an addition rate, H2 =Hga, to the atmosphere and an equal and opposite contribution, H1 = −Hga, to the land. Presumably these are meant to represent human-induced transfer of exchangeable carbon from the land 1

12 10 8 6 4 2 0 1750 1800 1850 1900 1950 2000 2050 2100 2150 2200 Figure 1: The annual, anthropogenic emission rates, Eσ , of carbon into the atmosphere versus calendar year since 1750, which is indexed by σ = 1.. 1 PgC is 1015 grams of carbon.

1 ppm of CO2 adds 2.18 PgC to the atmosphere.[2] to the atmosphere but no net addition of carbon to all four reservoirs. We have not included these terms in (3). They would imply that with no carbon at all at an initial time t = 0 (with |L⟩ = 0) the land would acquire negative quantities of carbon, dL1 /dt = −Hga. A straightforward way to model faster or slower exchanges between reservoirs j and k would be to increase or decrease the carbon capacitance Cj and Ck or the flow resistance Rjk , discussed in the capacitor model of Section 4 Fig. 1 shows an estimate[2] of annual human emissions of carbon, in the form of carbon dioxide since the year 1750. The gram molecular weight of CO2 and C are very nearly 44 and 12, so the rates of Fig. 1 must be multiplied by a factor of 44/12 = 3.67 to get the mass of emitted CO2 , the molecular form of almost all carbon produced by fossil fuel combustion and cement manfacture.

In Ed’s physics rate equations the natural exchange of carbon between the four reservoirs is described by a transfer-rate matrix     Γ11 Γ12 Γ13 Γ14 1/Tg −1/2Ta 0 0  Γ21 Γ22 Γ23 Γ24   −1/Tg 1/Ta −1/2Ts 0    . Γ= (4)  Γ31 Γ32 Γ33 Γ34  =  0 −1/2Ta 1/Ts −1/Td  Γ41 Γ42 Γ43 Γ44 0 0 −1/2Ts 1/Td Ed has parameterized his matrix Γ with four “e-times,” which he lists (in years) in Tables 3 and Table 4 of his paper: Tg = 25.0,

Ta = 3.22,

Ts = 4.90,

Td = 402.0.

(5)

As can be see from Ed’s assumed form of the transfer-rate matrix Γ, on the right of (4), half of the carbon leaving the atmosphere is assumed to go to the land and half to the shallow 2

ocean, an equal “split.” Similarly, half the carbon leaving the shallow ocean is assumed to go to the atmosphere and half to the deep ocean. It seems improbable that there would be exactly equal splits, but we will go along with the assumption. Unequal splits occur naturally in the capacitor model, also described by (1), that we will discuss later.

We conclude this introduction with a review of the relationship between the two most common measures of atmospheric carbon, the mass Mc of carbon in the atmosphere, or the fraction fc of CO2 molecules in the atmosphere. CO2 molecules are well mixed with other atmospheric molecules, except near the Earth’s surface, where respiration and photosynthensis of the biosphere can cause huge variations of fc from day to night. We therefore approximate fc as constant throughout the atmosphere. Recall that the mean surface pressure, which is slightly smaller than mean sea-level pressure because of higher altitude land surfaces, is p0 = g

dM = 0.985 × 106 dynes cm−2 , dA

(6)

comes from the weight of the air, that is from dM/dA, the mass per unit area of the atmosphere. We assume an altitude-independent acceleration of gravity g = 981 cm s−2 .

(7)

Most of the pressure (6) due to the nitrogen and oxygen molecules, but a small part is due to the carbon in CO2 molecules. The contribution of carbon to the pressure (6) is very nearly ∆p0 = fc p0

dMc mc =g . ma dA

(8)

where the gram molecular masses, mc of carbon atoms and ma of dry air molecules are mc = 12.01 gr

and

ma = 28.9647 gr.

(9)

The mean radius of the Earth is r = 6371 km = 6.371 × 108 cm.

(10)

We can multiply dMc /dA from (8) with Earth’s surface area, A = 4πre2 to find that the mass of carbon in the atmosphere is Mc = fc

4πr2 p0 mc = fc × 2.12 PgC ppm−1 . gma

(11)

So fc = 1 ppm of CO2 corresonds to Mc = 2.12 PgC of carbon.

2

Relaxation Modes

Ed numerically integrated the four coupled differential equations represented by (1) to find solutions under various scenarios. There are major advantages, both conceptual and in computational efficiency, to soving Ed’s physics rate equations with relaxation modes, instead of numerical integration.

3

2.1

Reservoir basis vectors

We introduce reservoir basis vectors |k⟩, defined for k = 1, 2, 3, 4, as the column vectors         0 0 0 1  0   0   1   0         (12) |1⟩ =   0  , |2⟩ =  0  , |3⟩ =  1  , |4⟩ =  0  .

1 0 0 0 The Hermitian conjugates of (12) are ⟨1| ⟨2| ⟨3| ⟨4|

= = = =

|1⟩† |2⟩† |4⟩† |4⟩†

= [1 = [0 = [0 = [0

0 0 0], 1 0 0], 0 1 0], 0 0 1].

(13)

The reservoir basis vectors have the orthogonality relations ⟨j|k⟩ = δjk ,

(14)

They also have the completeness relation, 

1 ∑  0 |k⟩⟨k| = 1 =   0 k 0

0 1 0 0

0 0 1 0

 0 0  .

0  1

(15)

Here k = 1, 2, 3, 4 in the sum of (15) and in similar expressions below. Using (13) and (2) – (4) we see that the amount of carbon in the kth reservoir, its anthropogenic fill rate and the transfer-rate matrix can be written as Lk = ⟨k|L⟩,

Hk = ⟨k|H⟩

and Γjk = ⟨j|Γ|k⟩.

(16)

For future discussions it is useful to define a column vector , |U ⟩, with unit elements , ⟨k|U ⟩ = 1, and its hermitian conjugate ⟨U |,   1  1  †  (17) |U ⟩ =   1  , and ⟨U | = |U ⟩ = [1 1 1 1 1].

1

2.2

Relaxation-mode basis vectors

By using relaxation-mode basis vectors, |ϕm ), instead of the reservoir basis vectors |k⟩ of (12), the four differential equations represented by the formal expression (1) can be “uncoupled.”

The uncoupled differential equations can be solved analytically, with no need for numerical

4

integration. The relaxation-mode basis vectors are the right eigenvectors, |ϕm ), of Γ. They are defined, aside from normalization, by ∑ Γ|ϕm ) = γm |ϕm ), or Γik ϕkm = γm ϕim , where ϕim = ⟨i|ϕm ).

(18) k

Here and in similar expressions below, the sum extends over the labels k = 1, 2, 3, 4 of the reservoirs. The symbol ϕim can be interpreted as the amplitude of the right eigenvector |ϕm ) for the carbon reservoir i or simply as the element of the 4 × 4 matrix ϕ. The eigenvalue corresponding to |ϕm ) is γm . The right eigenvector can be written as the column vector.   ϕ1m ∑  ϕ2m   |ϕm ) = |k⟩⟨k|ϕm ) =  (19)  ϕ3m  . k ϕ4m Later, we will show that the eigenvalues of Γ are non-negative, real numbers, γm ≥ 0. The elements ϕkm of the right eigenvectors can therefore be chosen to be real numbers. In general, Γ is not symmetric, so unlike the reservoir basis vectors |k⟩ of (12), the relaxation-mode basis vectors |ϕm ) need not be mutually orthogonal. They are analogous to the direct-lattice basis vectors of a crystal, which can be non-orthogonal for crystals with low symmetry. In analogy to (18), the left eigenvectors of Γ can be defined by ∑ {ϕn |k⟩Γkj = γn {ϕn |j⟩.

(20) {ϕn |Γ = {ϕn |γn , or k

The four eigenvalues, γm , defined by (18) or (20) will be identical. But unlike (13), the left eigenvectors will not be hermitian conjugates of the right eigenvectors, {ϕm | ̸= |ϕm )† .

To find the elements of the left eigenvector {ϕm |, we multiply (18) by ϕ−1 ni , an element of the inverse matrix ϕ−1 , and we sum on i to find ∑ ∑ ϕ−1 Γ ϕ = γ ϕ−1 (21) ij jm m ni ni ϕim . = γm δnm ij

i

We multiply both sides of (21) by ϕ−1 mk and sum over the mode labels m = 1, 2, 3, 4, to find ∑ ∑ ∑ −1 −1 −1 ϕ−1 Γ ϕ ϕ = ϕ Γ = γm δnm ϕ−1 (22) ij jm ik ni ni mk mk = ϕnk γn . ijm

m

i

Equating the second and fourth expressions in (22) gives (20), provided that we take the elements of the left eigenvector to be {ϕn |k⟩ = ϕ−1 nk , or {ϕn | = [ϕ−1 n1

ϕ−1 n2

ϕ−1 n3

ϕ−1 n4 ].

(23)

The left eigenvectors {ϕn | are analogous to the reciprocal-lattice vectors of a crystal of low symmetry, for which the direct-lattice vectors |ϕm ) are not orthogonal.

As defined by (18) and (23), the right and left eigenvectors have the orthogonality relations analogous to (14) {ϕn |ϕm ) = δnm .

(24) 5

They have a completeness relation analogous to (15),  1 0 0 ∑  0 1 0 |ϕm ){ϕm | = 1 =   0 0 1 m 0 0 0 ∑ Multiplying Γ on the left and right by m |ϕm ){ϕm | = 1, that the transfer-rate matrix can be written as ∑ Γ= γm |ϕm ){ϕm |.

 0 0  .

0  1

(25)

and using (18) and (24) we find (26)

m

∑ Multiplying (2) on the left by m |ϕm ){ϕm | = 1 of (25), ,we find the expansion of |L⟩ on the relaxation-mode basis vectors ∑ ∑ |L⟩ = |ϕm ){ϕm |L⟩ = |ϕm )L̃m , (27) m

m

where the amplitude for the mth relaxation mode is ∑ L̃m = {ϕm |L⟩ = ϕ−1 mk Lk .

(28)

k

Multiplying (28) by ϕjm and summing on m we find the inverse relation ∑ ϕjm L̃m .

Lj = ⟨j|L⟩ =

(29)

m

Substituting (27) into (1) we see that it leads to four uncoupled differential equations, one for each mode m, d L̃m = −γm L̃m + H̃m .

(30) dt The solution of (30) for times t > 0 is L̃m = e−γm t L̃m (0) + F̃m ,

(31)

where the human “forcing” of the mth mode is ∫ t ′ ′ F̃m = dt′ e−γm (t−t ) H̃m .

(32)

0 ′ Here H̃m = {ϕm |H(t′ )⟩, defined in analogy to (28), is the growth rate of the amplitude of the mode m due to human emissions of carbon at time t′ . Substituting (31) into (29), we find that the carbon level in the kth reservoir at time t > 0 is given by ∑ Lk = ϕkm L̃m m

=

∑

ϕkm L̃m (0)e−γm t + Fk ,

m

6

(33)

where the human forcing of the kth reservoir is ∑ Fk = ϕkm F̃m .

(34)

m

Eq. (33), the sum of an initial transient and a contribution Fk from human forcing, is the exact solution of Ed’s Physics Rate Equations. It is also the exact solution of the “capacitor” model that we will discuss below.

2.3

Basis-vector details

We denote the total amount of exchangeable carbon in the four reservoirs by ∑ L̂ = Lj = ⟨U |L⟩.

(35)

j

where |U ⟩ was defined by (16). The total rate of anthropogenic addition of carbon to all the reservoirs is ∑ Ĥ = Hj = ⟨U |H⟩.

(36) j

Using (1) with (35) and (36) we find that the growth rate of exchangeable carbon is ∑ d Γjk Lk + Ĥ.

L̂ = − dt jk

(37)

The transfer of carbon between reservoirs should not change L̂, the total amount of carbon in all reservoirs. That is, (37) should be independent of Γ,and should have the form d L̂ = Ĥ. dt Eqs. (37) and (38) will always be consistent if ∑ Γjk = 0, or ⟨U |Γ = 0,

(38)

(39)

j

where the unit row vector ⟨U | was given by (17). From (39) we see that each column of Γ must sum to zero. The constraint (39) means that carbon naturally removed from one reservoir flows to the other three. Ed’s matrix Γ of (4) has been constructed to satisfy the constraint (39).

It will be convenient to interpret (39) as the eigenvalue equation {ϕ4 |Γ = {ϕ4 |γ4 .

(40)

Comparing (39), (40) with (23) we see that {ϕ4 | = ⟨U |,

and

γ4 = 0 7

or {ϕ4 |k⟩ = 1 = ϕ−1 4k .

(41)

From (19), (24) and (41) we see that {ϕ4 |ϕ4 ) = 1 = ⟨U |ϕ4 ) = ϕ14 + ϕ24 + ϕ34 + ϕ44 .

(42)

Later, we will show that the ϕk4 must be non-negative real numbers, ϕk4 ≥ 0. So (42) can be interpreted to mean that ϕk4 = the equilibrium fraction of carbon in reservoir k.

(43)

Writing out the components of the eigenvalue equation (18) for m = 4 and γ4 = 0, and using Ed’s transfer-rate matrix (4), we find     0 ϕ14 /Tg − ϕ24 /2Ta  0   −ϕ14 /Tg + ϕ24 /Ta − ϕ34 /2Ts     Γ|ϕ4 ) = 0 =  (44)  0  =  −ϕ24 /2Ta + ϕ34 /Ts − ϕ44 /Td  .

0 −ϕ34 /2Ts + ϕ44 /Td The equations of (44) can be solved to find     ϕ14 Tg    ϕ24   = N  2Ta  . |ϕ4 ) =   2Ts   ϕ34  ϕ44 Td ∑ To ensure that j ϕj4 = 1 we see that the normalization factor N must be given by N = (Tg + 2Ta + 2Ts + Td)−1 .

Using the numerical values of Ed’s of carbon in the four reservoirs are    ϕ14 5.64  ϕ24   1.45     ϕ34  =  2.21 ϕ44 90.70

(45)

(46)

e-times from (5), we find that the equilibrium fractions    %, 

 compared to Ed’s

 5.6  1.5     2.2  %.

90.7

(47)

Shown on the right are the equilibrium perecentages from Ed’s Table 3, which he rounded to the nearest tenth of a percent. The eqilibrium percentages are the same to this level of precision.

Unlike the mode |ϕ4 ), for which the eigenvalue is γ4 = 0, the relaxing modes, |ϕm ) with m = 1, 2, 3, have positive eigenvalues, γm > 0, and the corresponding decay times are finite, −1 γm < ∞. There is no easy way to find closed-form expression like (45) for |ϕ1 ), |ϕ2 ) and |ϕ3 ).

But they can be determined by numerically solving the eigenvalue equation (18). This is fast and easy with modern mathematical software like MATLAB or Mathematica, but it may not be possible with Excel. Using the numerical values of Ed’s e-times from (5) to evaluate the transfer-rate matrix Γ of (4), and numerically determining the non-zero eigenvalues γm , we find the time constants  −1    γ1 2.462  γ2−1  =  7.218  yr.

(48) −1 γ3 80.523 8

250 Total 200

150

Atmosphere Deep Ocean Land

100

50

0 0

Shallow Ocean

10

20

30

40

50

60

70

80

90

100

Figure 2: Transfer of C atoms, initially all in the atmosphere, to the land and ocean, as described by (33) with Fk = 0 (no human emissions). At time t = 0, 212 PgC is assumed to be in the atmosphere with none in the land, shallow or deep oceans. The transients are from the solution of the physics rate equation (1) with no human emissions, |H⟩ = 0 and with the relaxation-mode formulas (33). These results are indistinguishable from those of those Ed’s Figure 19, which were calculated by numerical integration of the four coupled rate equations represented by (1).

The corresponding right eigenvectors |ϕm ) are        ϕ11 −0.4240 ϕ12 −0.6841  ϕ21   1.0000   ϕ22   0.4341         ϕ31  =  −0.7708  ,  ϕ32  =  1.0000 ϕ41 0.1948 ϕ42 −0.7500

  , 



   ϕ13 −0.7665  ϕ23   −0.1362       ϕ33  =  −0.0973  . ϕ43 1.0000

(49)

The vectors (49) have been normalized to make the largest amplitude 1. Carbon in the atmosphere makes the largest contribution to |ϕ1 ). Carbon in the shallow ocean makes the largest contribution to |ϕ2 ) and carbon in the deep ocean makes the largest contribution to |ϕ3 ).

We can use (41) and the orthogonality relation (24) to write ∑ {ϕ4 |ϕm ) = ϕkm = δm4 .

(50) k

One can verify that the amplitudes of the relaxing eigenvectors of (49) with m = 1, 2, 3 sum to zero as required by (50). According to (42) the elements of ϕk4 sum to unity. The left

9

35 30 25 20 15 10 5 0 1900

1920

1940

1960

1980

2000

2020

2040

2060

2080

2100

Figure 3: Predicted growth of atmospheric carbon due to human emissions. The carbon distribution between the land, atmosphere, shallow and deep oceans is assumed to have been in equilibrium in the year 1750, indexed by σ = 1. The annual emissions Eσ of Fig. 1, during years indexed by σ = 1, 2, 3, . . . , ρ, were used with (67) to calcuclate F2ρ , the atmospheric carbon fraction due to human emissions at the end of the ρth year. For the unrealisitic emission scenario of Fig. 1, no further carbon is added to the atmosphere after the year 2020, indexed by ρ = 270. Thereafter, the atmospheric carbon fraction starts to decrease due to transfer to the land and ocean. These results are indistinguishable from those of Ed’s Figure 13. The observed increase of atmospheric carbon has been much larger, about 130 ppm from the year 1750 to 2020. eigenvectors that correspond to (49), the first three rows of the inverse matrix ϕ−1 mk , are {ϕ1 | = [−0.0760 {ϕ2 | = [−0.1807 {ϕ3 | = [−1.0277

0.6957 0.4452 − 0.7086

− 0.3524 0.6739 − 0.3329

0.0022], − 0.0123], 0.0834].

(51)

3 Human Emissions An observational estimate of the time dependence of human emissions was shown in Fig. 1.

We have assumed that humans add carbon directly to the atmosphere, but not to the land or ocean. The elements of the human addition rate (3) can therefore be written as, Hk = δk2 E,

10

(52)

1

0.5

Observed Annual Cumulative

0

-0.5 1900

1920

1940

1960

1980

2000

2020

Figure 4: The ratio, f2ρ , predicted from (70), of the annual increase of carbon atoms in the atmosphere at the end of year ρ to the annual human emissions, assuming that only human emissions add to atmospheric carbon. For some years of the early 20th century, the predicted changes of atmospheric carbon shown in Fig. 2 were negative because more carbon was being absorbed by the land and oceans than was emitted by humans. Over the last few decades, the annual increase of atmospheric carbon is predicted to have been about 10% of the human emissions. The observed increase of atmospheric C has been about 50% of human emissions. with H2 = E. Using (52) in (32) we find that the human forcing of the mth mode can be written as ˜{m} , F̃m = ϕ−1 (53) m2 I where the mode integral is ˜{m}

I

∫

t

=

′

dt′ e−γm (t−t ) E ′ .

(54)

0

Here E ′ = E(t′ ). Substituting (53) into (34), we find that the contribution of human emissions to the carbon in the k reservoir is ∑ ˜ Fk = ϕkm ϕ−1 (55) m2 Im . m

Fig. 3 shows the growth of atmospheric carbon described by (55) for the emissions scenario of Fig. 1. It is indistinguishable from Ed’s Figure 13.

The rate of change of (55) is ( ) dFk ∑ ˜m , = ϕkm ϕ−1 E − γ I m m2 dt m 11

(56)

where we noted that

dI˜m = E − γm I˜m . dt The total growth rate of the carbon in all four reservoirs is dF̂ dt

=

∑ dFk

=

dt

k

∑

(57)

( ) {m} ˜ E − γ I ϕkm ϕ−1 m m2

km

(58)

= E.

One can use (50) and (41) to get the second line of (58) from the first. The ratio of anthropogenic carbon increase in reservoir k to that in all reservoirs is the ratio of (56) to (58), fk

3.1

) ( )−1 dFk dF̂ = = dt dt dF̂ ( ) ∑ ˜{m} γ I m = ϕkm ϕ−1 . m2 1 − E m (

dFk

(59)

Discretization

The data base[2] used to generate Fig. 1 consists of the 270 non-zero annual emissions, Eσ , for the calendar years ending in 1750, 1751, . . . , 2019. The year index is σ = 1, 2, . . . , 270.

Ed has extended the data base to the end of the year 2200 by appending zero emissions, Eσ = 0 for σ = 271, 272, . . . , 451. The total emissions are ∑ Eσ ∆t = 451.5540 PgC. (60) σ

The time interval of the data base is ∆t = 1 yr.

(61)

We can assume that the emission rate E = Eσ can be taken as constant during the σth calendar year, ending at the time tσ = 1749 yr + σ ∆t.

(62)

Denote the mode integral (54) at the time t = tρ by I˜ρ{m} = I˜{m} (tρ ).

(63)

Then we can write the mode integral as I˜ρ{m} =

ρ ∫ ∑ σ=1

tσ

′ −γm (tρ −t′ )

dt e

tσ −∆t

Eσ =

ρ ∑ σ=1

12

Ã{m} ρσ Eσ .

(64)

The elements of the vectors of (64) correspond to calendar years. The response matrix is a lower triangular matrix with the elements { {m} w ∆t e−γm ∆t(ρ−σ) , if ρ ≥ σ, {m} Ãρσ = (65) 0 otherwise.

The weights

{ w

{m}

=

(1 − e−γm ∆t )/(γm ∆t) 1

if if

m = 1, 2, 3, m = 4.

(66)

account for the mode decay during the integration interval ∆t. All of the elements of Ã{m} above the main diagonal are zero, and all the elements on the subdiagonal δ, with δ = ρ − σ, are equal to each other.

Substituting (64) into (55) we find ∑ {m} Fkρ = ϕkm ϕ−1 (67) m2 Ãρσ Eσ mσ

In analogy to (56) we define the annual growth of carbon in the kth reservoir for year ρ = 2, 3, 4, . . . by ∑ {m} ∆Fkρ = Fkρ − Fk,ρ−1 = ϕkm ϕ−1 (68) m2 ∆Ãρσ Eσ , mσ

The differential response matrix, {m}

{m} ∆Ã{m} ρσ = Ãρσ − Ãρ−1,σ ,

(69)

is defined for the indices ρ = 2, 3, 4, . . . , 451 and σ = 1, 2, 3, . . . , 451, with the number of columns exceeding the number of rows by 1. In analogy to (59), we define the fraction of annual emissions, Eρ in the year ρ that remains in the reservoir k by fkρ =

∆Fkρ , Eρ

(70)

with ∆Fkρ given by (68). The cumulative inventory fraction is Fkρ Êρ where the cumulative emissions is Êρ =

(71)

,

ρ ∑

Eσ .

(72)

σ=1

4

Capacitor Model

Ed’s Fig. 5 is based on Fig. 6.1 in Carbon and Other Biochemical Cycles, Chapter 6 of an IPCC report [3]. The IPCC report estimates levels of carbon in the reservoirs discussed

13

H1 L1 C1

R12 V1

H2

R23

H3

R34

H4

L2 V2

L3 V3

L4 V4

C2

C3

C4

Figure 5: Circuit model of the four carbon reservoirs. The carbon “capacities” of the land, atmosphere, shallow and deep oceans are C1 , C2 , C3 and C4 , the carbon contents are L1 , L2 , L3 and L4 and the “carbon potentials” are V1 , V2 , V3 and V4 . Human activities inject carbon currents of H1 , H2 , H3 and H4 into the reservoirs. The resistance to the flow of carbon between reservoirs j and k is Rjk = Yjk−1 , where Yjk is the flow admittance. above. The equilibrium reservoir contents before human addition of carbon were   2, 300  589   PgC. |L(0)⟩ =   900  37, 100

(73)

Ed has the same estimate in his Fig. 5 except that he rounded 589 to 590. In (73) and similar numerical estimates below, we have used three significant figures to facilitate comparisons of different model calculations. The numbers are probably uncertain by at least 10%.

One way to interpret the IPCC figures is with the circuit model shown Fig. 5. The kth reservoir has a capacitance Ck , measured in units of PgC cpu−1 . The “carbon potential,” Vk , of the kth reservoir is measured in “carbon potential units,” cpu. The resistance, Rjk , to carbon flow between reservoirs j and k is measured in units of cp yr PgC−1 . We have assumed no flow between reservoirs 1 and 3, 1 and 4, or 2 and 4, which means that R13 = ∞ R14 = ∞, and R24 = ∞. We assume that the carbon content Lk and carbon potenital Vk of the kth reservoirs are related by Lk = Ck Vk ,

or

|L⟩ = C|V ⟩.

(74)

The reservoir carbon content vector |L⟩ was defined by (2) and the “voltage” vector is defined

14

in like manner as

 V1  V2   |V ⟩ =   V3  .

V4 The diagonal capacitance matrix C can be written as  C1 0 0 0  0 C2 0 0 C=  0 0 C3 0 0 0 0 C4 The inverse of (74) is



|V ⟩ = C −1 |L⟩,

(75)

  . 

(76)

(77)

where C −1 is the diagonal matrix with elements, C1−1 , C2−1 , C3−1 , C4−1 .

Define the “carbon potential unit” as the common potential , 1 cpu = Vk (0), of the reservoirs before substantial human emissions of CO2 began. Then (74) implies that the reservoir capacitances are   2, 300 Lk (0)  589   PgC cpu−1 .

Ck = = (78)  900  Vk (0) 37, 100 The IPCC seems to have assumed that in equilibrium, before substantial human emissions of carbon dioxide, the carbon current, Ijk from reservoir j to reservoir k was equal and opposite the flow Ikj from reservoir k to reservoir j. This is much like the flow of electrons between dissimilar metals, j and k which can have very different conduction electron densities but which will have equal and opposite electron currents acrosss an unbiased junction. For carbon reservoirs, we do not need to be concerned about the “contact potentials” that are established between dissimilar metals by electron flow. Assuming that the slightly different forward and reverse flows shown in IPCC Fig. 6.1 are careless “roundoff error,” we average them to find the equilibrium currents     |I12 | 108  |I23 |  =  60  PgC yr−1 .

(79) |I34 | 102 We assume that the currents |Ijk | of (79) are those that would be driven though the “resistor,” Rjk by a potential difference of 1 cpu. Then the resistance Rjk to carbon flow between reservoirs j and k is 1 cpu 1 = , (80) Rjk = Yjk |Ijk | where Yjk is the admittance from flow between reservoir j and k. From (80) and (81) we see that the numerical values of the admittances are     Y12 108  Y23  =  60  PgC yr −1 cpu −1 .

(81) Y34 102 15

The most questionable part of this discussion is the numerical values assumed in (81) for flow admittances. As we will discuss below, secular increases atmospheric CO2 levels over the past century or so, as well as the decay of atmospheric 13 C from atmospheric tests of nuclear weapons are in rough agreement with predictions of the capacitor model if the atmosphereto-land and atmosphere-to-shallow-ocean admittances, Y12 and Y23 , are ten times smaller than the values given in (81.

For the situation disussed by the IPCC, there is no flow from reservoir 1 to 3, from 1 to 4, or from 2 to 4. So Y13 = Y14 = Y24 = 0. But for the time being, we assume that the only constraints on the admittances from reservoir j to another reservoir k are that the Yjk are symmetric and non-negative, Yjk = Ykj ≥ 0,

for

j ̸= k.

(82)

This means that any non-zero flow of carbon is always “downhill” from a reservoir with high carbon potential to one with lower carbon potential.

Using “Ohm’s law” we find that the capacitor charging rates are d L1 dt d L2 dt d L3 dt d L4 dt In vector notation, we

= (V2 − V1 )Y21 + (V3 − V1 )Y31 + (V4 − V1 )Y41 + H1 , = (V1 − V2 )Y12 + (V3 − V2 )Y32 + (V4 − V2 )Y42 + H2 , = (V1 − V3 )Y13 + (V2 − V3 )Y23 + (V4 − V3 )Y43 + H3 , = (V1 − V4 )Y14 + (V2 − V4 )Y24 + (V3 − V4 )Y34 + H4 .

(83)

can write (83) as d |L⟩ = Y |V ⟩ + |H⟩. dt

The admittance matrix is



Y11  Y21 Y =  Y31 Y41

Y12 Y22 Y32 Y42

Y13 Y23 Y33 Y43

The diagonal elements of the admittance matrix are ∑ Yjk Yjj = −

 Y14 Y24  .

Y34  Y44

(84)

(85)

(86)

k̸=j

The definition (86), together with (82), ensure that the elements of any row or of any column of the symmetric matrix Y sum to 0. ∑ ∑ Yjk = Yjk = 0 or Y |U ⟩ = 0, and ⟨U |Y = 0.

(87) j

k

The unit row vector ⟨U | and unit column vector |U ⟩ were given by ( 17). We can substitute (77) into (84) to find the physics rate equation (1), where the damping matrix is Γ = −Y C −1 .

16

(88)

The matrix Γ of (88) has all the properies of the relaxation matrices that we discussed in earlier sections. For example, because of (87), the elements of any column of the matrix (88) sum to zero, as required by (39). However, the special form of (88), the product of a real, symmetric admittance matrix Y with a diagonal, inverse-capacitance matrix C −1 , simplifies the relaxaxation-mode analysis and allows us to prove some important assertions of the previous sections.

5

Symmetrization

We introduce scaled versions of the reservoir carbon levels |L⟩ and the carbon addition rate |H⟩. |M ⟩ = C −1/2 |L⟩,

and |J⟩ = C −1/2 |H⟩. −1/2

−1/2

(89)

−1/2

−1/2

Here C −1/2 is the diagonal matrix with elements C1 , C2 , C3 and C4 −1/2 both sides of (1) by C and using the expression (88) for Γ, we find

. Multiplying

d |M ⟩ = −G|M ⟩ + |J⟩. dt

(90)

The scaled transfer-rate matrix, G = −C −1/2 Y C −1/2 ,

−1/2

or Gjk = −Cj

−1/2

Yjk Ck

,

(91)

is real and symmetric. Therefore G will have four real eignenvalues γm and four orthogonal eigenvectors |θm ⟩.

G|θm ⟩ = γm |θm ⟩.

(92) Mathematical software is more likely to have routines to find eigenvalues and eigenvectors of symmetric matrices like G, than for matrices like Γ of (88) which are not symmetric unless all reservoir capacitances are identical, C1 = C2 = C3 = C4 .

The eigenvectors |θm ⟩ of (92) can be chosen with orthonormality properties anlogous to (14) and (15) of the reservoir basis vectors |k⟩, ⟨θn |θm ⟩ = δnm , and

∑

|θm ⟩⟨θm | = 1,

(93) (94)

m

where

⟨θm | = |θm ⟩† .

(95)

Multipling (92) on the left by ⟨θm | and using (93) we see that the eigenvetors G are given by γm = ⟨θm |G|θm ⟩.

(96)

Multipying both sides of (92) by C 1/2 and using (91) and (91), we find we find ΓC 1/2 |θm ⟩ = γm C 1/2 |θm ⟩.

17

(97)

1

0.8

0.6

0.4

0.2

0 0

10

20

30

40

50

60

70

80

90

100

Figure 6: Transfer of C atoms, initially all in the atmosphere, to the land and ocean, as described by (33). The assumptions are nearly the same as for Fig. 2 or Ed’s Figure 19, but we used the capacitor model where 64% of atmospheric carbon losses are to the land, and 36% to the shallow ocean. Since Fig. 2 assumed 50% from the atmosphere to both the land and to the shallow ocean, it showed a slightly slower transfer of carbon to the land and a slightly faster transfer to the shallow ocean. The initial pulse was normalized to 1 PgC, so the vertical scale can also be interpreted as the fraction of carbon in each of the four reservoirs. Only about 14% of the airborne fraction remains after 10 years. After the atmospheric nuclear tests in the 1950’s a bit more than 50% of 14 C remained airborne after ten years [1], judging from Ed’s Fig. 2.

Thus, to within an arbitrary, real normalization coefficient Nm , the right eigenvectors of Γ are |ϕm ) = Nm C 1/2 |θm ⟩.

(98) The matrices G and Γ have the same eigenvalues, γm , but different eigenvectors. From (98) we can write |θn ⟩ = Nn−1 C −1/2 |ϕn ) and ⟨θn | = (ϕn |C −1/2 Nn−1 .

(99) Here we used the hermitian-conjugagte relations (ϕm | = |ϕm )† and (C −1/2 )† = C −1/2 . Substituting (99) and (98) into (93) we find −1 ⟨θn |θm ⟩ = Nn−1 Nm (ϕn |C −1 |ϕm ) = δnm

(100)

This will be consistent with the orthonormality relation (24) if we let the left eigenvector of Γ be {ϕn | = Nn−2 (ϕn |C −1 = Nn−1 ⟨θn |C −1/2 .

(101)

18

35 30 25 20 15 10 5 0 1900

1920

1940

1960

1980

2000

2020

2040

2060

2080

2100

Figure 7: Predicted growth of atmospheric carbon due to human emissions. The calculations were done in the same way as for Fig. 3, except that the capacitor model with the parameters of (78) and (81) were used. Slightly larger atmosperic accumulations are predicted, a peak of about 34 ppm versus 32 ppm. The large discrepancy with the observed accumulation of about 130 ppm remains.

6

Non-Negative Eigenvalues

In this section we prove that for transfer-rate matrixes Γ of the form (88), the eigenvalues γm are non-negative γm ≥ 0.

(102) Negative eigenvalues would cause time-evolution factors to diverge for large times, e−γm t → ∞ for t → ∞ if γm < 0.

Consider an arbitrary, real column vector |g⟩ and its hermitian conjugate ⟨g|   g1  g2  †  |g⟩ =  (103)  g3  , and ⟨g| = |g⟩ = [g1 g2 g3 g4 ]. g4 The “expectation value” of G with respect to |g⟩ is ∑ −1/2 −1/2 ⟨g|G|g⟩ = − gj Cj Yjk Ck gk jk

=

∑

[( )2 ( )( ) ( )2 ] −1/2 −1/2 −1/2 −1/2 Yjk gj Cj − 2 gj Cj gk Ck + gk C k

j<k

=

∑

( Yjk

−1/2 gj Cj

−

−1/2 gk Ck

j<k

19

)2 ,

(104)

1

0.5

Observed Annual Cumulative

0

-0.5 1900

1920

1940

1960

1980

2000

2020

Figure 8: Annual and cumulative airborne fractions of carbon as for Fig. 4, except that the capacitor model with the parameters of (78) and (81) were used. As for Fig. 4 the predicted airborne fractions are much smaller than the observed fractions, which are closer to 50%. or ⟨g|G|g⟩ ≥ 0.

(105)

The first line of (104) consists an algebraic sum of 24 terms, each proportional to one of the six possible admittances Yjk , for indices j < k. Recall that Ykj = Yjk ≥ 0 and note that (86) can be used to write the diagonal elements Yjj in terms of the Yjk with j ̸= k. Letting |g⟩ = |θm ⟩ in (105) and using (96) we find (102).

Of special interest is the vector |g⟩ = |c⟩, where  1/2  C1  1/2    C (106) |c⟩ = C 1/2 |U ⟩ =  21/2  ,  C3  1/2 C4 The elements of |c⟩ were chosen to make the expression (104) for ⟨c|G|c⟩ equal to zero. Using (106) with (91) and (87) we see that G|c⟩ = −C −1/2 Y C −1/2 C 1/2 |U ⟩ = C −1/2 Y |U ⟩ = 0.

(107)

So |c⟩ is proportional to the eigenvector |θ4 ⟩ of G with eigenvalue γ4 = 0, and we can write |θ4 ⟩ =

|c⟩ , (Tr C)1/2

(108)

where the trace of the capacitance matrix is Tr C = C1 + C2 + C2 + C4 .

20

(109)

Choosing the normalization coefficient to be N4 =

1 , ( Tr C)1/2

(110)

we can use (98) to write the non-relaxaing right eigenvector of Γ as   C1  C|U ⟩ 1   C2  , |ϕ4 ) = = Tr C Tr C  C3  C4

(111)

From (101) we find that the non-relaxing left eigenvector of Γ is {ϕ4 | = N4−2 (ϕ4 |C −1 = ⟨U |,

(112)

in agreement with (41).

7

Numerical Results of the Capacitor Model

Using relaxation modes to analyze the time evolution of carbon in the four reservoirs with the capacitance model of Fig. 5, we find results that differ little from those obtained earlier with Ed’s “physics rate equation.” For example, the equilibrium reservoir fractions are     ϕ14 5.63  ϕ24   1.44      (113)  ϕ34  =  2.20  %. ϕ44 90.73 The percentages of (113) are almost identical to those given by (47) for Ed’s model. The the time constants for the three relaxing modes are  −1    γ1 2.834  γ2−1  =  6.629  yr.

(114) γ3−1 89.088 These are quite similar to the time constants (48) relaxing right eigenvectors |ϕm ) are        ϕ11 −0.5994 ϕ12 −0.5416  ϕ21   1.0000   ϕ22   0.3069         ϕ31  =  −0.5923  ,  ϕ32  =  1.0000 ϕ41 0.1917 ϕ42 −0.7653

of Ed’s physics rate equations. The   , 



   ϕ13 −0.7743  ϕ23   −0.1509       ϕ33  =  −0.0748  . (115) ϕ43 1.0000

The vectors are differ little from those of Ed’s model, given by (49).

21

120 100 80 60 40 20 0 1900

1920

1940

1960

1980

2000

2020

2040

2060

2080

2100

Figure 9: Predicted growth of atmospheric carbon due to human emissions. The calculations were done in the same way as for Fig. 7, except that the smaller flow rates implied by (119) were used. This large increase of the flow resisistance brings the predicted increase from human emissions close to the observed increase of about 130 ppm.

8

Physics Rate Equations Versus Capacitor Model

Ed’s physics rate equations are a special case of the capacitor model. As summarized by Ed’s Fig. 7, his physics rate model is equivalent to a capacitor model with slightly different reservoir capacities than those implied by the carbon inventories of IPCC Fig. 6.1 prior to the year 1750. That is, instead of (78), Ed’s model has   2, 307  594   PgC cpu−1 .

Ck =  (116)  904  37, 095 Instead of (81) Ed’s Fig. 7 implies that     Y12 92.3  Y23  =  92.3  PgC yr Y34 92.3

22

−1

cpu

−1

.

(117)

1

Annual Observed 0.5

Cumulative

0

-0.5 1900

1920

1940

1960

1980

2000

2020

Figure 10: Annual and cummulative airborne fractions of carbon as for Fig. 8, but with the much smaller flow rates implied by (119). This brings predicted fractions very close to observed fractions.

All of the non-zero admittances have the same value, Y12 = Y23 = Y34 = Yp = 92.3 PgC yr−1 cpu−1 .

Ed’s e-times are C1 Yp C2 Ta = 2Yp C3 Ts = 2Yp C4 Td = Yp Tg =

= 24.92 yr = 3.19 yr = 4.88 yr = 401.95 yr (118)

We don’t think that Ed makes a very persuasive case for using his physics model versus the capacitance model. However, the final results are quite similar for both models, as one can see by comparing Fig. 7 with Fig. 3, or by comparing Fig.6 with Fig. 2. So this is not a big issue. Predicted increases of atmospheric carbon are much smaller than observed increases.

Perhaps the models seriously overestimate the carbon flow from the atmosphere to the land and oceans. Or perhaps the emissions from fossil-fuel combustion and cement manufacture, shown in Fig. 1, are not the only source of additional atmospheric carbon. Or there could be some combination of these factors.

23

1

0.8

0.6

0.4

0.2

0 0

10

20

30

40

50

60

70

80

90

100

Figure 11: Transfer of C atoms, initially in the atmosphere, to the land and ocean, as described by (33). The assumptions are the same as for Fig. 6, but with the much smaller flow rates implied by (119). Some 75% of the carbon is predicted to remain airborne after ten years. After the atmospheric nuclear tests in the 1950’s a bit more than 50% of 14 C remained airborne after ten years [1], judging from Ed’s Fig. 2.

9

Are IPCC Flow Rates too Large?

As outlined above, if we we try to use the equilibrium flow rates estimated from IPCC Figure 6.1 for the year 1750 to determine the flow of carbon to the land and ocean, the carbon flows out of the atmosphere to the land and oceans too quickly for human emissions to account for the observed increase of atmospheric carbon dioxide. To see what happens, we tried increasing the resistance to flow out of the atmosphere by a factor of 10 for the capacitor model, so that (81) becomes     Y12 108/10  Y23  =  60/10  PgC yr −1 cpu −1 .

(119) Y34 102 We have kept the reservoir capacities (78) the same. As shown Fig. 9 and Fig. 10, increasing the flow resistance by a factor of 10 brings the predicted atmospheric carbon increase from human emissions fairly close to the observed increases. The transfer of atmospheric carbon to the other three reservoirs slows down substantially, as shown in Fig. 11. The observed loss 14 C in the years after the atmosperic tests of nuclear weapons might still be marginally consistent with the smaller admittances (119), but they can’t be made much smaller.

24

100

80

60

40

20

0 0

10

20

30

40

50

60

70

80

90

100

Figure 12: Comparison of the decrease of carbon in the atmosphere according to (120) of the physics model, for an emission pulse E ′ = E(t′ ) = L0 δ(t′ ) and the analogous Bern-model formula (124), with L0 = 100. The graph can hardly be distinguished from Ed’s Figure 18.

10

The Bern Model

In notes with the title, The IPCC Bern Model, dated March 9, 2020, Ed Berry gives a useful summary of the IPCC Bern Model, which describes the transfer of carbon from the atmosphere to the land and oceans. We will refer to these notes as BM. Equation (1) of BM will be called (BM1), with analogous notation for subsequent equations.

In (BM2), Ed writes a formula for the amount of carbon remaining in the atmoshere at a time t after an amount L0 of carbon is suddenly injected at time t = 0, L(t) = L0 [A0 + A1 exp(−t/T1 ) + A2 exp(−t/T2 ) + A3 exp(−t/T3 )] .

(120)

The carbon remaining is the sum of a constant, steady-state term, L0 A0 and three terms of amplitudes L0 AS , which decay exponetially with time constants TS , for S = 1, 2, 3.

The basic integral equation (BM1) for the Bern model has the same form as the exact solution (33) of Ed’s physics rate equations, or the exact solution for the capacitor model.

Combining (54) and (55) we find, Fk =

4 ∑ m=1

ϕkm ϕ−1 m2

∫

t

′

dt′ e−γm (t−t ) E ′ .

(121)

0

where our E ′ = E(t′ ) plays the same role as BM’s ECO2 (t′ ). The indices, S = 0, 1, 2, 3 in BM, are related to our mode indices, m = 1, 2, 3, 4, by m = 4 − S. 25

(122)

The symbol ρCO2 on the left side of (BM1) corresponds to F2 of (121), the atmospheric level of carbon at time t.. If we substitute E ′ = L0 δ(t′ )

(123)

into (121) to represent a pulsed addition of L0 units of carbon at time t′ = 0, and carry out the integral for k = 2, the index of the atmospheric reservoir, we find [ ] −γ1 t −γ2 t −γ3 t F2 = L0 ϕ21 ϕ−1 + ϕ22 ϕ−1 + ϕ23 ϕ−1 + ϕ24 ϕ−1 (124) 12 e 22 e 32 e 42 .

Here we noted that e−γ4 t = 1 since γ4 = 0. From inspection of (124) and (120) we see that the Bern fractions, fCO2 ,S = AS must be ϕ2m ϕ−1 m2 = fCO2 ,S = AS .

We therefore have

4 ∑

ϕ2m ϕ−1 m2

=

m=1

3 ∑

fCO2 ,S =

S=0

3 ∑

(125)

(126)

AS = 1.

S=0

−1 γm

The mode time constants of (48) for Ed’s physics rate equations correspond to atmospheric exponential decay times τCO2 ,S = TS of the Bern model, 

   γ1−1 2.462  γ2−1   7.218       γ3−1  =  80.523  yr γ4−1 ∞ The corresponding fractions are     ϕ21 ϕ−1 0.6957 21  ϕ22 ϕ−1   0.1933  22     −1  ϕ23 ϕ32  =  0.0965  ϕ24 ϕ−1 0.0145 42

 ←→

  τCO2 ,3 T3  τCO2 ,2   T2     τCO2 ,1  =  T1 τCO2 ,0 T0   fCO2 ,3 A3  fCO2 ,2   A2     fCO2 ,1  =  A1 fCO2 ,0 A0 

←→





 2.57   18.0  =    171  yr. ∞ 

(127)



 0.316   0.279  =    0.253  .

0.152

(128)

We have plotted the atmospheric carbon transients of (120) and (121) in Fig. 12. The results cannot be distinguished from those of Figure 18 of Ed’s paper[1]. Although Ed calculated the transient by numerically integrating the physics rate equations, it is also conveiently described as the sum of a constant term and three decaying exponentials, just like the Bernmodel transient. Ed already mentioned that the Bern model assumes an absurdly large atmospheric steady-state of 15.2 % of the total carbon.

11 Atmospheric Growth of CO2 and Depletion of O2 Fig. 13 shows the changes of atmospheric CO2 and O2 measured by the Scripps O2 , CO2 and APO Observatory in Alert, Canada [5]. Over the period from 1990 to 2015, CO2 levels increased at nearly the same rate, about 1.9 ppm yr−1 . O2 depletion rates accelerated from 26

−600

350

Carbon dioxide [ppm] 360 370 380 390

−400 −200 Oxygen/Nitrogen [per meg]

400

0

Carbon Dioxide and Oxygen in Atmosphere

1990

1995

2000

2005

2010

2015

Year Carbon dioxide [ppm]

Oxygen/Nitrogen [per meg]

Measurements at Alert, Nunavut, Canada

Figure 13: Measured levels of atmospheric CO2 and O2 from the Scripps O2 , CO2 and APO Observatory in Alert, Canada ( 82◦ north latitude). For the period shown here, the average growth rate of atmospheric CO2 was nearaly constant at 1.9 ppm/yr. Oxygen levels decreased at an accelerating rate, from 2.5 ppm/yr in the early 1990’s to 4.5 ppm/year after 2010 (4.77 per meg= 1 ppm). about −2.5 ppm yr−1 just after 1990 to −4.0 ppm yr−1 just before 2015. The marked annual oscillations are due to preponderance of photosynthesis over respiration in the spring and summer, which releases O2 and depletes CO2 , and the preponderance of respiration over photosynthesis in the fall and winter. The fall-winter increase of CO2 is about 50 ppm yr−1 or 106 PgC yr−1 , with a similar spring-summer decrease. It is hard to draw any quantitative conclusions from this data alone. They are qualitatively what one would expect.

12 Law Dome Ice Cores Another important series of observational data are the past atmospheric CO2 levels in ice cores from the Law Dome on Antarctica. These show that since the year 1006 until about 1850, the inferred atmospheric levels averaged about 280 ppm, with a decrease of around 5 ppm to around 275 ppm just before the year 1600, perhaps signalling the beginning of the Little Ice Age, and with a subsequent increase of about 5 ppm just around the year 1800, perhaps due to its end. Systematic large increases of atmospheric CO2 did not begin until after 1850, which is consistent with human emissions shown in Fig. 1.

27

Figure 14: Measured CO2 fractions from ice cores drilled at the Law Dome in Antarctica[6].

The fact that the fraction was nearly constant from the year 1006 to about 1850 is consistent with the CO2 levels in Ed’s four reservoirs being in equilibrium during that time. The relatively small observed changes may have been relatated to the Medieval Warm Period and the Little Ice Age.

13 Summary We conclude this review of Ed Berry’s paper with a summary of our views.

1. Unlike many others, Ed is very clear about his mathematical assumptions. He writes down crisp, linear rate equations for the flow of carbon between four reservoirs, the land, the atmosphere, the shallow ocean and the deep ocean.

2. IPCC’s original Bern model was probably based on rate equations similar to Ed’s. But a natural question is whether it is possible, in principle, to model the carbon cycle with linear rate equations. Carbon exchanges between the atmosphere, land and oceans are very complicated and involve a great deal of biology. The equations of fluid flow for the atmosphere and oceans (e.g. Navier-Stokes) are also famously nonlinear.

3. Ed’s rate equations have an equilibrium distribution of carbon between the reservoirs.

Without human emissions, the equilibrium distribuion does not change with time. In Ed’s equilibrium, about 1.45% of carbon is airborne, 90.70% is in the deep ocean, 2.21% is in the shallow ocean and 5.64% is on the land.

4. The existence of an unperturbed equilibrium distribution of carbon is consistent with the CO2 fractions measured in air bubbles trapped in ice cores taken from the Law 28

Dome in Antarctica[6], shown in Fig. 14. The atmospheric fraction has remained close to 280 ppm from about the year 1000 to 1850.

5. Ed’s rate equations conserve carbon. If there are no human emissions, carbon lost from one reservoir flows to the other three so the total carbon content of all reservcoirs remains constant. If there are human emissions, they equal the total increase of carbon in all four reservoirs.

6. As we have pointed out earlier in the review, normal relaxation modes are a more efficient way to solve Ed’s rate equations than the numerical integration he used. But Ed does not make mistakes, in numerical integration.

7. The IPCC is very vague about what the numbers it cites really mean. It will be difficult to write Ed’s paper in a way that demonstrates that the IPCC is mistaken in its estimates of carbon inventories and flows. IPCC can simply say that whatever Ed (or anyone else) writes down is a straw-man argument, criticizing equations that IPCC never wrote down. But the IPCC’s “Bern Model” looks like it started from rate equations similar to Ed’s.

8. The IPCC can say that the transfer rates that Ed assumed between the reservoirs are much too big. As we showed in Section 9, one can fix most of the problems that Ed identified by assuming that the real flow rates from the atmosphere to the land and ocean are ten times smaller than those inferred from IPCC cartoons. With ten times smaller flow “admittances,” about half of emissions would remain in the atmosphere, and the time for removing a pulse of atmospheric carbon would be reasonably close to what was observed for atmospheric tests of nuclear weapons.

9. The treatment of the all-important oceans in Ed’s paper is probably too sketchy. Ed does not discuss the biological pump that can carry carbon quickly through the shallow ocean to the deep ocean. Airborne carbon can be transferred directly into the deep ocean throgh the formation of cold, salty, “carbonated” water near the poles. The release of carbon into the atmosphere involves upwelling of deep ocean water that was formed centuries ago.

10. If Ed’s rate equations are correct some other source of airborne carbon is needed. Ed is not very clear about what this other source could be, but the two largest reservoirs are the deep oean and the land. It is hard to understand why, after some 800 years of apparent equilibrium (as implied by Fig. 14), the deep ocean might start to outgas CO2 more rapidly around the year 1850.

11. An increase of land emissons due to human activities sounds more reasonable than an increase of ocean emissions. Carbon from respiration of living things on the land would be depleted in the heavy isotope 13 C, compared to the dominant light isotope 12 C. So land emissions would be consistent with the observed “lightening”[7] of the airborne fraction of CO2 . But it is not clear that more carbon is being released from the land.

In the US, east of the Mississipi river, the land has almost certainly been a major sink for carbon because of the regrowth of forest on land that can no longer be profitably farmed.

29

References [1] E. X. Berry, Preprint, March 7, 2020, The Core Issues of the Human Carbon Cycle.

[2] Historical Emissions of Carbon Dioxide, https://cdiac.ess-dive.lbl.gov/trends/ emis/glo_2010.html [3] IPCC, Carbon and Other Biogeochemical Cycles,https://www.ipcc.ch/site/assets/ uploads/2018/02/WG1AR5_Chapter06_FINAL.pdf [4] E. X. Berry, Preprint, March 9, 2020, The IPCC Bern Model.

[5] SCRIPPS O2, CO2 AND APO, climate-data/scripps-o2-co2-and-apo

https://climatedataguide.ucar.edu/

[6] D. M. Etheridge et al., Historical Records from the Law Dome DE08, DE08-2 and DSS Ice Cores, https://cdiac.ess-dive.lbl.gov/trends/co2/lawdome.html [7]

13

C/12 C isotope mixing.html

ratios,

https://www.esrl.noaa.gov/gmd/outreach/isotopes/

30