1
Supplementary Information:
Part A. This section presents the derivation of Eqs 1-3 in the text. References are listed in the
main text.
This derivation is based on the following assumptions: steady state, infinitely fast mixing of the
solid phase, constant porosity, constant parameters (not depth dependent), ignoring explicit
treatment of dissolved carbonate speciation, linear dissolution kinetics (justified in Fig. 1), and
ignoring explicit treatment of the effects of oxic organic matter decomposition.
The following equations govern dissolved and solid carbonate, as well as the “Clay” component
of the solids. In the sediment, a) for dissolved CO32-
d 2C
j Dc 2 + (1- j ) kc ( Cs - C ) B = 0
dx
(A1)
where x (m) is the depth into the sediment, with origin at the sediment-water interface and
positive downward into the sediment. The first term of this equation accounts for diffusion in
the porewaters, while the second term represents the source of dissolved carbonate from CaCO3
dissolution into the porewaters. Papers in the geochemical/oceanographic literature normally use
a different form, that can be shown to be equivalent, i.e.,
kc' (1- W)b = kc' rCaCO3 (1-
C
b
k'
)
= c rCaCO3 (Cs - C)B = kc (Cs - C)B
Cs rCaCO3 Cs
(A2)
where kc' is a different rate constant and b is the CaCO3 mole concentration per unit volume of
solids.
b) Calcite
FB - j Dc
dC
= (1- j )wrCaCO3 B
dx
(A3)
c) “Clay” (non-carbonate fraction of the solids)
FM = (1- j )wm
(A4)
where m is the mass concentration of “Clay” in the solids (kg m-3) and FM is the flux of “Clay” to
the sediment-water interface (kg m-2 y-1).
d) Solid Volume Fraction
2
B+
m
rM
=1
(A5)
where r M is the mass density of the “Clay” (kg m-3).
Boundary conditions on this system of equations are:
-j Dc
dC
dx
= b ( C ¥ - C0 )
(A6)
x=0
where C0 is the dissolved carbonate ion concentration at the sediment-water interface. At depth
in the sediment,
-j Dc
dC
dx
=0
(A7)
x®¥
The solution to Eq. A1 is
C(x) = a1 e-
kx
+ a2 e
kx
(A8)
where a1, a2 are integration constants and
k=
(1- j )kc B
j Dc
(A9)
Equation A7 immediately gives that
a2 = 0
(A10)
Equation A6 next gives that
a1 =
b (C¥ - C0 )
j Dc k
(A11)
Next, the distribution of the solute in the diffusive boundary layer is linear,
C(x) = a3 x + a4
(A12)
where a3 and a4 are constants.
The conditions on a3 and a4 are that C(0) = a4 and C(-δ) = C∞, where x = -δ is the top of the
(diffusive) boundary layer. The second condition gives
3
a3 = -
( C ¥ - C0 )
(A13)
d
By matching Eqs. A8 and A12 at x = 0, there results an equation for the unknown concentration
at the sediment-water interface,
b C¥
j Dc k
C0 =
b
1+
j Dc k
Cs +
(A14)
B is obtained from Eq. A3; specifically, substitute Eq. A6 to arrive at
FB + b ( C¥ - C0 ) = (1- j )wrCaCO3 B
(A15)
or with Eq. A14,
FB - k * ( C¥ - Cs ) = (1- j )wrCaCO3 B
(A16)
which is Eq. 1 of the text, and where
k* =
ks b
ks + b
(A17)
and
ks = [j (1- j )kc Dc B ]
1/2
(A18)
The burial speed w is obtained by combining and manipulating Eqs. A4, A5 and A16,
é FB - k * ( C¥ - Cs ) ùû
FM
w= ë
+
(1- j )rCaCO3
(1- j ) r M
(A19)
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Part - B. Intercepts can occur in experiments like those performed by Keir (1983) due to a
number of effects. There is no a priori reason to believe that a rate plot must go through the
origin. Why?
First and foremost, Keir did not attempt replicate experiments to assess the precision of this data,
nor did he use a true standard to obtain an estimate of the accuracy. Therefore there are
unreported experimental errors that might account for the intercepts. (This statement does not
constitute a criticism of Keir’s work. His experiment was both inspired and difficult; it still sets
the standard today.)
In addition, Keir measured the saturation state in the overlying water, not in the porewater, and
there is usually a difference. The source of that difference lies in the presence of other
simultaneous reactions. The sediment used in Keir’s study is natural and, geochemically
speaking, not a pure system, but contains other reactive materials, i.e., organic matter and
possibly other carbonate materials, that undergo reactions during the calcite dissolution
experiments, and these can affect the dissolved carbonate species in the porewaters.
Specifically, oxic organic matter decomposition (given that the sediments are typical deep sea
sediments) will produce CO2, i.e.,
CH2O + O2  CO2 + H2O
(B1)
where CH2O is stoichiometric organic matter, and thus force dissolution within the sediment, to
some extent, even when the overlying water supersaturated. This was established by Emerson
and Bender (1982) and confirmed by many other studies thereafter.
This process may not dominate dissolution, but it can cause intercepts in plots such as Fig. 1.
The effect of this reaction on the porewaters would be to make them more under-saturated than
expected, i.e., the overlying waters may be super-saturated, but the CO2 generated by reaction
(B1) can make the porewaters less saturated, even under-saturated. If the overlying waters are
under-saturated, reaction (B1) can make the porewaters even more under-saturated. A plot of
overlying saturation (1-Ω) versus measured rate of dissolution would then not have a zero-zero
intercept because the porewaters could be under-saturated when (1-Ω) = 0 in the overlying
waters.
Another reaction is the dissolution of aragonite, in addition to calcite. Aragonite can be present,
and Keir (1983) thinks there is trace to minor aragonite in the 117GGC sample. Aragonite is
more soluble than calcite, and dissolution of this additional mineral in the sediments would cause
the porewaters to me more saturated than expected by calcite dissolution alone. Aragonite
dissolution could suppress calcite dissolution relative to that suggested by the calcite undersaturation in the overlying waters and produce a positive intercept on a Rate-versus-Saturation
plot. That may be occurring in some samples.
Furthermore, Keir (1983) appears to have used a solubility value that is too large, a point raised
previously by Hales and Emerson (1997). That means that the (1-Ω) values in the original paper
are too small, and consequently there will be a positive intercept on a Rate-versus-Solubility plot.
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This is particularly true of the dissolution experiments with the pure calcite beds. Using a
solubility that is too large does not, however, affect the rate constant values because the slope is
unchanged by this error, i.e., the values of Cs does not determine k*,
k * = slope =
¶Rate
Cs ¶(1- W)
R2 - R1
R - R1
= 2
Cs [ (1- W2 ) - (1- W2 )] [ C1 - C2 ]
(B2)
where Cs is calcite solubility, R2 and R1 are rates at saturation Ω2 and Ω1, respectively, and C2
and C1 are the CO32- concentrations that correspond to Ω2 and Ω1, respectively. (Note Ω2 =
C2/Cs, etc.) The rate constants are thus correct.
Beyond these experimental factors, assuming that the origin, i.e., R = 0, (1-Ω) = 0, is a priori
included in any experiment introduces a non-justified assumption and grants privileged status to
that point. The rate at (1-Ω) = 0 is a measurable quantity. Because it is measurable, the value
measured at that point needs to be used in modeling the data (along with its associated
uncertainty). Keir did not measure the rate precisely at saturation and rigorous statistical
treatment of this experimental data would argue that we cannot substitute our biased, anticipated
value for the missing datum. As there are other reactions that affect the carbonate system in the
sediment, as argued above, one cannot assume that R = 0 in the sediment when (1-Ω) = 0 in the
overlying water.
Finally in interpreting this data, I invoke the Principle of Parsimony, a.k.a. Occum’s razor. What
is the simplest, but realistic, explanation to the data. Overall the data are, at a glance, linear.
Would an intelligent, but unbiased, observer use a non-linear rate law? Only if told that the
origin must be included. But the above argues that introducing such a restriction is statistically
biased, and there are geochemical reasons to expect non-zero intercepts.
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Part – C. The steady state results given above can be generalized to the case of transient ocean
acidification. The response time of the boundary layer and the sediment porewater system to a
transient change in overlying water concentration, C∞(t), is quick compared to the time-scale of
acid penetration in the oceans, i.e., the concentration profile in the boundary layer can adjust in
less than an hour, the sediment system in a few hours to weeks, while the time scale for
acidification is decades to millennia. Therefore, the boundary layer and sediment porewater
system can me treated as if they are in a steady state, and most of the equations derived above
can be used with C∞ as time variable.
The time dependence does, however, appear in the equation for B. We must take into account
the change in B with time due to added dissolution. Thus, Eq. A17 reads as
dB
1
éë FB - k * ( C¥ (t) - Cs ) - (1- j )wrCaCO3 B ùû
=
dt rCaCO3
(C1)
where t is time and all other terms are defined as before. The change on the left-hand side refers
to the inventory of CaCO3 above the “erodible surface” [see Archer et al., 1998], i.e., the CaCO3
in the sediment that can be mixed upward and dissolved at the sediment-water interface. The
thickness of that layer can be calculated as given in Archer et al. [1998].
The burial velocity w is a function of time in this equation; it can be calculated from timedependent conservation of “Clay”,
dm
= FM - (1- j )w m
dt
(C2)
and Eq. A5; i.e., substitute Eq. A5 into Eq. C2 to obtain
-
dB
1
=
é FM - (1- j )wr M (1- B ) ùû
dt r M ë
(C3)
If we now add Eqs. C1 and C3, there results an algebraic equation for w,
éë FB - k * ( C¥ (t) - Cs ) ùû
FM
w=
+
(1- j )rCaCO3
(1- j )r M
(C4)
Equation C1 is of the same form as that used in Boudreau et al. [2010a], but differs from Zeebe
[2012] in that it explicitly allows for boundary layer transport control, as argued in the text.