Sharp decrease in long-term chemical weathering rates along an altitudinal transect
Clifford S. Riebe*
Department of Earth and Planetary Science, University of California, Berkeley, CA 94720-4767,
riebe@seismo.berkeley.edu, 510-643-2171
James W. Kirchner
Department of Earth and Planetary Science, University of California, Berkeley, CA 94720-4767,
kirchner@seismo.berkeley.edu, 510-643-8559
Robert C. Finkel
Center for Accelerator Mass Spectrometry, Lawrence Livermore National Laboratory, Livermore, CA
94551, and Department of Earth Science, University of California, Riverside, CA 92521
finkel1@llnl.gov, 925-422-2044
*corresponding author
Electronic appendix
Here we provide a detailed description of our methods for quantifying long-term rates of
denudation and chemical weathering from cosmogenic nuclides and geochemical mass balance. We also
include additional information about our sampling approach.
1. Mass balance approaches for measuring long-term chemical weathering rates
1.1. Theory
Long-term chemical weathering rates have typically been inferred using a mass balance approach
[1,2], in which chemical weathering outputs are inferred from the changes wrought in parent material as it
is converted to weathered soil. In the mass balance approach, chemical weathering losses are estimated
from measurements of immobile element enrichment in the weathered material. Elements that are
immobile during chemical weathering become enriched as other elements are removed by dissolution; the
greater the mass lost through dissolution, the greater the relative enrichment of the immobile elements
that are left behind.
1
1.2. Chemical weathering rates from non-eroding soils of known age
If physical erosion from a soil has been negligible, as may be the case on some flat-topped
moraines [3] and fluvial terraces [4], mass losses can be attributed to chemical weathering alone. If so,
measurements of immobile element enrichment yield estimates of mass loss that can be interpreted as
chemical weathering rates using soil ages, assuming they can be quantified. Soil ages provide rate
constants for averaging total mass losses from weathering over time, thus yielding long-term average
chemical weathering rates. However, because erosion must be negligible, this approach is not useful for
determining the extent to which rates of physical erosion and chemical weathering are interrelated.
Hence, although rates of fresh mineral supply [e.g., 5,6] and thus physical erosion [7] are thought to
significantly affect chemical weathering rates, the linkage between them has remained speculative in the
absence of co-located measurements of long-term rates of chemical weathering and physical erosion.
Moreover, because non-eroding soils of known age are rare, conventional soil mass balance methods
cannot be widely applied, and have yielded few measurements of long-term chemical weathering rates.
1.3. Chemical weathering rates from eroding landscapes
In mountainous settings, most soils have substantial physical erosion rates. In soils undergoing
steady-state erosion, the enrichment of immobile elements can still provide a measure of the total mass
lost due to chemical weathering, just as it does in a non-eroding soil. But in eroding landscapes, soil age
is difficult to define, because soil is continually renewed as fresh rock is incorporated from below,
replacing weathered soil removed by physical erosion at the surface. In soils undergoing such
denudation, cosmogenic nuclides can be used to measure rates of denudation and chemical weathering, as
we show below [8-10].
2
1.3.1. The relationship between immobile element enrichment and chemical weathering rate for eroding
soils
For a soil undergoing steady-state formation, erosion, and weathering (such that the mass of
weathered material in storage on the landscape is approximately constant through time), conservation of
mass (Fig. 3A) implies that the rate of conversion from parent material to weathered products will be
equal to the total denudation rate, which in turn will equal the sum of the rates of chemical weathering
and physical erosion:

R=D=E+W
(A1)
where R is the rate of conversion of rock to weathered material, D is the total denudation flux, E is the
flux of weathered material removed by physical erosion, and W is the chemical weathering flux, all in
mass per area per time [8]. Each of the terms in Eq. A1 is a mass flux; therefore the densities of rock and
soil do not explicitly appear, nor are they necessary for the analysis that follows. Denudation rates are
often reported in the literature in units of length per time, reflecting rates of landscape lowering. These
can be straightforwardly scaled by parent material density to yield the denudation mass flux used in our
analysis. Note also that the cosmogenic nuclide literature often refers to total denudation rates -- which is
what cosmogenic nuclides measure -- as erosion rates [e.g., 9]. Here we use the term "physical erosion"
to distinguish it as the physical component of the total denudation flux.
Conservation-of-mass equations like Eq. A1 can also be written for individual elements of the
rock and soil:

D · [X]rock = E · [X]soil + WX
(A2)
where [X]rock and [X]soil are the concentrations in rock and soil of an element X, and WX is its chemical
weathering rate. For immobile elements, WX is zero (Fig. 3B) and Eq. A2 reduces to
3
 [Zr] rock 

E  D 
 [Zr] soil 

(A3)
where [Zr]rock and [Zr]soil are the concentrations in rock and soil of an immobile element, in this case
zirconium. Substituting Eq. A3 into Eq. A1 yields
 [Zr] rock 

W  D 1 [Zr]
soil 

(A4)
Equations A3 and A4 can be combined to yield a non-dimensional measure of weathering rates:
W  [Zr] rock 
  CDF
 1 D  [Zr]soil 
(A5)
where CDF, called the "chemical depletion fraction", expresses the fraction of total denudation that is
accounted for by chemical weathering.
The bulk chemical weathering rate of Eq. A4 can also be partitioned into the weathering rates of
individual elements. Substituting Eq. A3 into Eq. A2 directly yields,

[Zr] rock 

WX  D [X]rock - [X]soil
[Zr]
soil


(A6)
 [Zr] rock 

EX  E  [X] soil  D  [X] soil 
 [Zr] soil 
(A7)
and similarly,
4
where EX is the physical erosion rates of element X. Note that the weathering rate WX can be expressed in
non-dimensional form by normalizing it by the total denudation rate for element X, as follows:
 [X] soil [Zr] rock 
WX
  CDFX
 1 D  [X] rock  [X] rock [Zr] soil 
(A8)
Normalized weathering rates from Eq. A8 are similar to chemical depletion fractions of Eq. A5, except
they express the fraction of denudation accounted for by chemical weathering rates on an element-byelement basis, rather than for the soil as a whole. Hence we refer to them as "chemical depletion fractions
of individual elements" (and denote them with CDFX). Note that CDF's are chemical weathering rates
that are normalized by total denudation rates. Hence, they should be useful for accounting for mineral
supply effects on chemical weathering rates, with any differences in CDF's from site to site largely
reflecting effects of non-erosion-related factors, such as precipitation rate and temperature.
1.4. The importance of parent material homogeneity
All soil mass balance methods, including ours, assume that immobile element enrichment reflects
mass losses due to weathering. However, if soil is not generated from a single, uniform parent material,
its weathering enrichment will be difficult to quantify, because its bulk chemistry will reflect mixing of
multiple parent materials in addition to element depletion and enrichment due to weathering losses. For
example, if a soil is generated from two rock types, there are two inputs in Eq. A1 rather than one. Unless
the relative rates of soil production from each rock type can be determined, their relative contributions to
immobile element concentrations in the soil will be difficult to disentangle from the effects of weathering
enrichment. Hence, mass balance methods like those in Eqs. A4 and A6 are best applied where soils are
formed from a single parent material [8].
5
1.5. Steady-state soil depth
Equation A1 assumes that soil depth is in steady state. If this were not the case, then chemical
weathering rates inferred from Eqs. 4 and 5 would be in error. Eq. A1 would become
R = E + W + soil · dh/dt
(A9)
where dh/dt is the rate of change of soil depth h and soil is soil density. Hence, if the mass imbalance
between soil production and total denudation is small compared to the weathering rate, the error will be
small enough to ignore. Theoretical considerations and field observations suggest that this should be the
case at our sites. If soil production rates decrease with increasing soil thickness, as theory has predicted
[11,12] and as soil production measurements from hilly landscapes have shown [13], then soils should
maintain relatively stable depths over the long term, even if soil production and removal become
unbalanced from time to time [12]. For example, if the rate of soil loss (i.e., by physical erosion and
chemical weathering) decreases, soils will begin to thicken, causing soil production to decrease until it
again balances soil removal, thus stabilizing soil depth at a slightly thicker value. If soil removal rates
instead increased, soil depth would also stabilize, but at a slightly thinner value. Field observations
suggest that soil-forming processes at our sites are dominated by biogenic activity, which presumably
decreases in effectiveness with increasing soil thickness. This implies that soil production rates probably
vary inversely with depth, and that soil depth should therefore be roughly stable.
1.6. Zirconium as an immobile element
Results from recent laboratory experiments suggest that Zr may not be completely impervious to
chemical weathering [14]. However, field examples of Zr mobility are limited to lateritic soils [e.g., 15],
which are generally formed where chemical weathering rates are fast and physical erosion rates are slow,
such that intensely altered soils are not quickly removed and replenished with fresh rock from below.
Erosion rates are relatively fast at our sites compared to those of typical laterites, so exposure to intense
6
weathering in soils will be relatively short. Using a maximum soil thickness of 60 cm, a soil density of
1.3 g·cm-3, and our 117 t·km-2·yr-1 average denudation rates, we estimate that soil residence times are less
than 7 kyr. Hence, although we cannot rule out the possibility that some Zr is dissolved and lost from
some of our soils, Zr dissolution significant enough to affect our analysis seems unlikely.
1.7. Eolian fluxes
Our mass balance approach assumes that eolian fluxes have a negligible effect on soil bulk
chemistry. This will be the case where eolian fluxes are small compared to total denudation rates. This
should be true in our sites because they are all subject to denudation rates greater than about 100
t·km-2·yr-1, which is much higher than any plausible inputs or outputs due to eolian processes. Soil
formation continually supplies fresh material as physical erosion and chemical weathering remove altered
products from catchment hillslopes. If eolian fluxes were large compared to rates of soil formation, we
would observe evidence of eolian deposition and remobilization, but no such evidence is present at our
sites.
1.8. Results from the mass balance approach compared with results from three independent techniques
In previous work at Rio Icacos, Puerto Rico, we compared chemical weathering rates inferred
from cosmogenic nuclides and geochemical mass balance with results from conventional approaches for
measuring chemical weathering rates [8]. Rio Icacos is an ideal location for such a comparison because
chemical weathering rates have been inferred from three independent sets of data, spanning both short and
long timescales: stream solute fluxes from water samples collected at the Rio Icacos gauge [16,17], solute
concentrations and infiltration rates of regolith porewater [18,19], and bulk chemical losses from a
regolith profile [19]. Our estimates of weathering rates of Si, Na, Ca, Mg, and K agree closely with the
three independent sets of weathering rate data, thus confirming the accuracy of our mass balance
approach for measuring long-term chemical weathering rates.
7
2. Quantifying denudation rates with cosmogenic nuclides
The geochemical mass balance of Eqs. A1-8 yields chemical weathering and physical erosion
rates of soils and their component elements from measurements of immobile element enrichment,
concentrations of constituent elements in rock and soil, and total denudation rates. Denudational mass
flux rates (i.e., D in Eqs. A1-4, 6 and 7) can be measured, over timescales comparable to those of soil
formation, using cosmogenic nuclide methods. Because these methods have come into widespread use
only in the past few years, geochemical mass balance methods have only recently become widely
applicable to eroding landscapes [8-10].
10
Be is produced in quartz grains near the earth's surface by cosmic ray neutrons and muons [20].
Because quartz grains at depth are shielded from cosmic radiation, cosmogenic 10Be concentrations reflect
near-surface residence times of quartz, and can be used to infer long-term average rates of outcrop erosion
[20], landscape denudation [21-23], and soil production [13,24].
For quartz grains that have eroded steadily to the surface from great depth, the accumulation of
cosmogenic 10Be in quartz at depth z beneath the surface can be expressed as
dN ( z , t )
N ( z, t )
 Pn ( z )  Pμ  ( z )  Pμf ( z ) 
dt
τ
(A10)
where t is time,  is the radioactive meanlife of 10Be (2.18±0.09 Myr; after [25]), and Pn(z), P-(z), and
Pf(z) are the depth-dependent production rates by neutron spallation, negative muon capture, and fast
muon reactions. In this analysis, depth below the surface (z) is measured by the mass per unit area
overlying (and thus shielding) the sample. Thus z has units of mass per unit area.
Cosmogenic nuclide production by neutrons declines exponentially with depth below the surface
such that:
8
Pn(z) = Pn,0 ∙ e-z / 
(A11)
where Pn,0 is the spallogenic 10Be production rate at the surface (5.1 at∙g-1∙yr-1 at sea level, high latitude;
after [26]), and  is the penetration lengthscale for production by neutron spallation (160±10 g∙cm-2; after
[27]).
Stopped negative muons produce 10Be at a rate that can be modeled by
P- (z) = (z) ∙ Y
(A12)
where (z) is the negative muon stopping rate as a function of depth, z, and Y is the yield of 10Be per
stopped negative muon, equal to 5.6 ∙ 10-4 [28]. The stopping rate profile for negative muons in rock,
compiled by Stone et al. [29] and adapted for analytical integration of Eq. A1 by Granger and Smith [30],
is

(z) = A1 · e-z / L1 + A2 · e-z / L2)
(A13)
where A1 = 170.6, A2 = 36.75, L1 = 738.6 g·cm-2 and L2 = 2688 g·cm-2 at sea-level, high latitude.
Granger and Smith [30] combined experimental 10Be production rates determined from targets
irradiated by fast muons [28], along with fast muon energy and flux profiles [also from 28], to estimate
that 10Be production by fast muons can be approximated by
Pf(z) = B · e-z / L3
(A14)
with B = 0.026 and L3 = 4360 g·cm-2 at sea level, high latitude.
For mineral grains undergoing steady-state denudation at rate D, the mass overlying and shielding
the sample, and thus the ambient production rate, are both time-dependent, with dz/dt = -D. Using this
relationship, along with Eqs. A11-14, to solve Eq. A10 yields
9
N0 
Pn, 0
1 τ  D Λ 

Y · A1
Y ·A2
B


1 τ  D L1  1 τ  D L 2  1 τ  D L 3

(A15)
showing that nuclide concentrations in mineral grains that have eroded from great depth to the surface can
be expressed as a function of their steady-state denudation rate [31].
Equation A15 requires that the eroding quartz has no inherited 10Be, as should be the case for
most bedrock-derived soils in eroding, upland catchments; inheritance should only be important for soils
whose parent materials have had long histories of near-surface exposure, as may be the case on moraines
and alluvial fans and terraces.
2.1. Sediment mixing and radioactive decay of 10Be
Equation A6 can be used to determine the steady-state denudation rate of rock, but in the present
analysis we aim to measure denudation rates of catchments where quartz grains in the regolith may have
complicated histories of exposure and burial. Several researchers [21-23] have shown that cosmogenic
nuclide concentrations in well-mixed sediment can be used to infer the average erosion rate of the
sediment's source area, provided that erosion is sufficiently rapid that radioactive decay of the
accumulating nuclides can be ignored. If radioactive decay cannot be ignored, then denudation rates
inferred from Eq. A15 will overestimate the true denudation rate by inadequately representing the most
slowly eroding parts of the sediment contributing area [31]. For nucleon spallation products, Eq. A15
shows that radioactive decay will be negligible if D/>> 1/, or D >> 0.7 t·km-2·yr-1. Equation A15
further shows that negligible decay of products of interactions with fast muons requires denudation rates
to be significantly faster than 20 t·km-2·yr-1 (i.e., such that D >> L3/). Denudation rates at our sites are
significantly higher than that (Table 2, main text), indicating that radioactive decay should be minimal,
and is unlikely to introduce any biases in spatially averaged denudation rates inferred from average
nuclide concentrations in our sediments and soils. Even at the most slowly eroding sites, any such biases
10
should be insignificant, because 10Be production due to fast muons accounts for less than about 3 %, a
small fraction of the total.
2.2. Accounting for effects of quartz enrichment
Recent work has shown that soil dissolution may introduce an important bias to denudation rates
inferred from cosmogenic nuclides in soils [24], particularly if chemical weathering rates are rapid [32].
Quartz is a relatively insoluble component of soils, so it will have a relatively long soil residence time if
dissolution of more soluble minerals leads to significant quartz enrichment. We expect quartz enrichment
to be significant enough in many of the the extreme climates considered here that we need to account for
it in our cosmogenic nuclide measurements of denudation rates. Including the effects of prolonged
residence time due to quartz enrichment, Eq. A15 becomes,
N0 
Pn ,0
1 τ  D Λ 

· Fsoil Frock  1  Fsoil Frock  · e

ρsoilh Λ
Y ·A2
B

1 τ  D L 2  1 τ  D L 3
  1 τ Y ·DA L
1
1


(A16)
where soil and h are soil density and thickness, and Fsoil and Frock represent the fraction of quartz in the
soil and rock [24,31]. We assume that quartz and zircon are both effectively insoluble in groundwater,
and approximate the quartz enrichment ratio, Fsoil/Frock, with measurements of zirconium enrichment,
[Zr]soil/[Zr]rock [32]. It seems unlikely that weathering is intense enough that mass loss from quartz
dissolution at Santa Rosa Mountains should be any more than a negligible component of the total. Hence,
errors introduced by substituting [Zr]soil/[Zr]rock for Fsoil/Frock should lead to at worst, only slight
overestimation of D.
11
2.3. Latitude and altitude scaling of
10
Be production rates
All of the production rate constants listed above are given for sea level and high latitude. Due to
variations in geomagnetic and atmospheric shielding, production rates will vary from sample to sample,
depending on latitude and altitude [e.g., 33]. Whereas spallogenic production can be scaled with latitude
and altitude according to relationships derived for the cosmogenic nucleon flux (Table 2 of [20]), scaling
of muogenic production has not yet been standardized into such a simplified approach. To simplify our
analysis, we assume that latitude scaling of the muon flux is negligible; because latitude scaling depends
complexly on muon energy, including it explicitly in our calculations would require a significant increase
in calculation complexity, without much improvement in resolution of individual denudation rates. Errors
introduced by this simplification should be small, because cutoff rigidities at Santa Rosa Mountain are <5
GV [34], which has been cited as upper limit for latitudinal invariance of the muon flux [35].
We scale the 10Be production due to fast muons (i.e., B in Eqs. A14, A15 and A16) for the effects
of altitude using eP / L3, where P is the difference between atmospheric pressure at sea level and the
sampling locality. To scale 10Be production due to negative muon capture from sea level to our sites, we
use the muon flux profile relationship compiled by Stone et al. [29] for depth-dependent scaling of muons
in rock. This will only be approximately correct, because scaling of the muon flux in rock (as reported by
Stone et al. [29]) is largely unaffected by the energy-dependent radioactive decay of muons which is
known to occur in the atmosphere [36]. In rock, muons are attenuated mostly by interactions with
matter, because rock is sufficiently dense that attenuation by muon decay is small by comparison. In
contrast, muons spend enough time traversing the atmosphere that significant radioactive decay occurs
there. Muons of higher energies decay less, because they travel fast enough that they experience
relativistic time dilation, which effectively retards their decay relative to that of lower-energy muons.
Muons of energy >10 GeV undergo the least amount of decay, and scale with depth in the atmosphere
essentially as they do with depth in rock [31]. Hence by scaling negative muon production for all
energies using the relationships of Stone et al. [29] for rock, we are ignoring the relatively rapid
12
atmospheric attenuation of the lower-energy portion of the incident muon flux. We can verify that this
leads to only slight underestimation of denudation rates by simplifying to the other extreme, and scaling
negative muons according to eP / 247, as if all energies experienced decay as do low-energy muons, with
minimal time dilation effects (after [36]). When we do this, the resulting estimates of denudation rates
are at most only 10% higher than what we report in the main text. This indicates that the much more
complicated analysis, including energy-dependent decay, would change denudation rates reported here by
<10%, too little to warrant the added complexities of such an approach, given that absolute uncertainties
in production rates are ~20% (e.g., [20]).
3. Sampling rocks, saprolite and soils and measuring their bulk and trace element chemical
compositions
Measurements of the rock-to-soil enrichment of insoluble elements permit us to quantify
chemical weathering rates from estimates of total denudation rates (see Eqs. A1-A8). Our measurements
of denudation rates, inferred from cosmogenic nuclides in widely distributed soils and alluvial sediment,
are areal averages, so the measurements of soil and rock chemistry that we use to infer chemical
weathering rates should also be representative, areal averages. In other words, for consistency between
the cosmogenic nuclide and weathering depletion measurements, we need representative element
concentrations of pre-weathered rock and of weathered soil. Therefore, we sampled material from widely
distributed rock outcrops, soil surfaces, and soil pits, and, after measuring the bulk chemical composition
of each sample, we used catchment-wide averages of soil and rock composition for our weathering rate
calculations.
Soil surfaces were sampled semi-randomly: we divided our sampling locations into grids,
occupied the approximate grid point locations in the field and sampled soil surfaces from randomly
chosen 1-2 m2 areas. We also typically dug several soil pits, so that we could sample subsurface soil
material (i.e., colluvium) and saprolite (i.e., the chemically altered, but physically intact bedrock material
13
at the base of mobile colluvium). Soil pits were located at widely distributed points within each location,
in an effort to sample material from the widest possible range of conditions. These soil pits revealed that
the colluvium generally lacks strong vertical zonation.
Fresh outcrops were also generally chosen from widely distributed points in each locality, in an
effort to sample as much of any lithologic variability as possible. The average [Zr] of subsurface soil
material is, for the most part, very similar to that of soil on the surface, indicating that Zr concentrations
from surface samples alone are generally representative of the [Zr] of soils and can therefore be used to
infer representative weathering rates.
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