Rock-weathering rates as functions of time

QUATERNARY RESEARCH 15, 250-264 (1981)

Rock-Weathering Rates as Functions of Time

STEVEN M. COLMAN U.S. Geological Survey, Box 25046, Denver Federal Center, Denver, Colorado 80225

Received February 12, 1980

The scarcity of documented numerical relations between rock weathering and time has led to a common assumption that rates of weathering are linear. This assumption has been strengthened by studies that have calculated long-term average rates. However, little theoretical or empirical evidence exists to support linear rates for most chemical-weathering processes, with the exception of congruent dissolution processes. The few previous studies of rock-weathering rates that contain quantitative documentation of the relation between chemical weathering and time suggest that the rates of most weathering processes decrease with time. Recent studies of weathering rinds on basaltic and andesitic stones in glacial deposits in the western United States also clearly demon- strate that rock-weathering processes slow with time. Some weathering processes appear to con- form to exponential functions of time, such as the square-root time function for hydration of volcanic glass, which conforms to the theoretical predictions of diffusion kinetics. However, weathering of mineralogically heterogeneous rocks involves complex physical and chemical processes that generally can be expressed only empirically, commonly by way of logarithmic time functions. Incongruent dissolution and other weathering processes produce residues, which are commonly used as measures of weathering. These residues appear to slow movement of water to unaltered material and impede chemical transport away from it. I f weathering residues impede weathering processes then rates of weathering and rates of residue production are inversely proportional to some function of the residue thickness. This results in simple mathematical analogs for weathering that imply nonlinear time functions. The rate of weathering becomes constant only when an equilibrium thickness of the residue is reached. Because weathering residues are relatively stable chemically, and because physical removal of residues below the ground surface is slight, many weathering features require considerable time to reach constant rates of change. For weathering rinds on volcanic stones in the western United States, this time is at least 0.5 my.

INTRODUCTION

Although almost every text in geomor- phology, soils, and related fields includes some discussion of weathering rates, variation in rock-weathering rates through time is poorly documented. Few studies have provided sufficient qnantitative data on both time and weathering variables to es- tablish their functional relations.

Weathering rates provide useful estimates of time for stratigraphic and correlation studies, and for deciphering recent geologic history. In addition, weathering rates have “practical” application to as- sessment of recurrence intervals of floods, landslides, earthquakes, and volcanic eruptions. Evaluation of recurrence intervals of these hazards is greatly enhanced by use of numerical (mostly radiometric) age determinations, but in many cases numerical techniques are not applicable. In the ab-

sence of numerical ages, relative-age methods, such as soil development and rock weathering, provide the only means of es- timating hazard recurrence intervals.

The purposes of this paper are: (1) to ex- amine common assumptions about rates of chemical weathering that result from inade- quate time control; (2) to review studies that have documented numerical relations between rock weathering and time; (3) to provide an analysis of recent data from an extensive study of weathering rinds on volcanic clasts in deposits of various ages (Colman and Pierce, 1980); and (4) to ex- amine mathematical analogs for rates of development of weathering features.

PREVIOUS STUDIES OF RATES OF WEATHERING

Studies that have dealt with rates of chemical weathering can be divided into

250 0033-5894/81/030250-15$02.00/O Copyright 0 1981 by the University of Washington. All rights of reproduction in any form reserved.

ROCK-WEATHERING RATES 251

two classes by the degree to which they document the actual behavior of weathering rates through time. “Uncontrolled” studies are those in which adequate time control was not available, or in which weathering variables were not measured quantitatively. These studies are by far the most numerous and have led to a common assumption that rates of weathering are linear. The second, smaller class of studies can be called “controlled” studies. These studies have used numerical time control and quantitatively measured chemical weathering of rocks.

Uncontrolled Studies and the Linear Rate Assumption

Several types of uncontrolled studies, lacking the data to determine a functional relation between weathering and time, have suggested linear weathering rates, either as long-term averages, or as the simplest hy- pothesis in the absence of time calibration. Some studies have calculated rates of weathering from dissolved material re- moved by streamflow. These data provide excellent quantitative measures of present chemical weathering rates. Although these rates have been used as long-term averages, past variation in climate and in associated water budgets make it difficult to evaluate how these rates have changed with time. Many such studies are described by Loughnan (1969) and Oilier (1969).

Qualitative measurement of weathering variables, although useful for many purposes, cannot be used to define the functional relation between weathering and time. Most studies that measure weathering features qualitatively do so in terms of qualitative classes. As an example, Rahn (1971) studied the weathering of tombstones cut from various rock materials and obviously had good time control; however, only qualitative classes of weathering (unweathered, slightly weathered, exten- sively weathered) were described.

Other studies have examined material exposed to weathering for a known interval of time. These studies contain considerable

quantitative data for various weathering features and provide average rates of weathering for the time since the material began to weather. However, such average rates are by definition linear functions and provide little information concerning the actual relation between weathering and time. As an example, Hay (1960) provided extensive data on the alteration of a 4000- year-old volcanic ash. The rates of alteration calculated for the ash, however, are linear averages for the past 4000 years-the actual behavior of the weathering rate through time cannot be determined for the single ash.

For the purpose of defining the relation between weathering and time, lack of quantitative age control is the most serious problem in using the uncontrolled studies. Age sequences of deposits or materials ex- hibiting different degrees of weathering are not uncommon. However, without independent time control, these deposits or materials form only relative-age sequences in which the age sequence is defined by weathering variables.

Reviews of the relation between weathering and time, such as those by Oilier (1969), Birkeland (1974), and Winkler (1975) contain numerous examples of the types of uficontrolled studies discussed above. As a result of the scarcity of data on the functional relation between weathering and time, the relation commonly is assumed to be linear. Nelson (1954), studying the weathering of a sequence of glacial deposits in the Rocky Mountains, perhaps best summarized the dilemma:

“Certainly these functions are not strictly linear functions of time.” (p. 335)

But,

“For simplicity and from the lack of anything better, a11 processes are considered linear functions of time.” (p. 339)

Despite its simplicity in the absence of time control, a linear relation can probably be considered either fortuitous or the result of specific processes. Without time calibra-

252 STEVEN M. COLMAN

tion, no compelling reason exists for assuming that rates of weathering remain constant, rather than increasing or decreasing, through time.

Controlled Studies

Studies that have actually documented weathering rates by comparing quantitative measures of weathering with numerical estimates of time can be classified into three groups based on the type of time control used. The first group is experimental studies, in which the elapsed time of the experiment provides time control. These studies commonly accelerate naturally occurring weathering processes by raising temperature, increasing water flow, or decreasing grain size. The second group is studies of the weathering of man-made features such as tombstones and buildings, using historical records for time control. The third group is studies of weathering of naturally occurring deposits or materials for which the duration of weathering is known from independent age estimates. Geologic materials examined in these studies are as much as several million years old.

Experimental studies. Schaller and Vli- sidis (1959) observed the oxidation of a sample of powdered siderite for a period of 43 years. Their results show that the rate of oxidation of the siderite decreases with time (Fig. 1). The relation between the oxidation of the siderite and time is logarithmic; i.e., the rate of oxidation decreases exponentially. This relation fits the equation

FeO(%) = a + b log(1 + t)

where t = time (years), and a and b are constants.

Grant (1969) showed that abrasion pH, which measures capacity for hydrolysis, is a function of time (Fig. 2). The rate of change of abrasion pH decreases with time and quickly approaches zero as equilibrium with the surrounding water is reached. Hy- drolysis of silicates during natural, near- surface weathering occurs in an open system; equilibrium with weathering solutions

TIME, YEARS

FIG. 1. Oxidation (percent FeO) of siderite as a function of time. Data from Schaller and Vlasidis (1959). The linear least-squares best-fit to the data is: FeO(%) = 59.4 - 35.7 Iog(1 + t).

is probably rarely attained in such environments, with the result that the rate of hydrolysis would continuously decrease.

Correns and his colleagues pioneered ar- tificial weathering experiments in the laboratory, beginning in the 1930s (reviewed in Correns, 1963). Since then numerous experiments have been performed on a variety of minerals using several experimental methods. Notable examples include

I I

0 1 2 3 4 5 TIME, MINUTES

FIG. 2. Example of changes in abrasion pH with time. Data for a pulverized fresh granitic rock from Grant (1969).


McClelland (1950), Wollast (1967), Nickel (1973), Lagache (1965, 1976), Petrovic and others (1976), and Holdren and Berner, (1979). These experiments simulate weathering by passing a steady flow of water through fragments of minerals or by agitat- ing the fragments in a buffered closed system. Slow natural weathering rates are commonly accelerated by grinding minerals to increase surface area, by raising temperatures, or by increasing the flow of water. Two observations are common to virtually all these experiments, under a wide range of conditions. First, silicate minerals dissolve incongruently, that is, the amounts of species released into solution do not corre- spond to their stoichiometric proportions in the mineral. Second, the rate of release of species to the solution decreases with time (Fig. 3), eventually becoming constant in some cases.

Incongruent dissolution requires the formation of some type of residue. Many workers have postulated that the observed decrease in the rate of release of ions with time is due to the presence of a residual layer on dissolving mineral grains. Ions would have to diffuse through an increasing thickness of such a residual layer as the mineral dissolves. Helgeson (1971) used theoretical diffusion kinetics to develop parabolic (square root of time) rate functions for the release of ions into solutions, and used these functions to model the data from several of the weathering experiments. Constant rates of ion release observed in some experiments after a time can be explained if the dissolution becomes congruent, or if the residual layer reaches an equilibrium thickness, being destroyed at the layer-solution interface at the same rate it is formed at the layer-mineral interface. The composition of the residual layer has been variously described as a cation-depleted (leached) layer (for example, Paces, 1973), an amorphous precipitate (for example, Correns, 1963; Wollast, 1967), or a crystalline precipitate (for example, Helgeson, 1971). Nixon (1979)

-0 50 100 150

TIME. DAYS

2.5 6

0 Si

3 2.0

0 3 ij 1.5

p 1.0 ! A

5 10 15 20 25 30

TIME. DAYS

TIME, DAYS

FIG. 3. Results of experimental weathering of silicates, measured as species released into solution. (A) Potassium feldspar, data from Correns (1963): (BI Hornblende, data from Nickel (1973): (Cl Sanadine. selected data (surface area, 800 cm*/g; mass, 0.2 g) from Lagache (1976); curves are least-squares best-fit\ to a square-root time function.

suggested that different types of feldspars may have different types of residual layers.

However, several recent studies have questioned the existence of an impeding residual layer (Petrovic et al., 1976; Lagache, 1976; Berner, 1978; Bemer and Holdren, 1979, Holdren and Berner, 1979). From a


variety of observational data these studies suggest that a compact residual layer does not form on feldspars or other minerals and that surface reactions control the rate of mineral dissolution. Petrovic (1976) argued that the residual layer, if it exists, is too porous to impede dissolution, and, therefore, that the dissolution is surface-reaction controlled. Lagache (1976) modeled her experimental data with a square-root time function (Fig. 3), but argued against diffusion through a residual layer being respon- sible for that relation.

Surface-reaction-controlled dissolution should yield constant rates of release of ions into solution. Holdren and Berner (1979) explained the nonlinear rate of dissolution observed in nearly all experiments by a two-stage process. Initially, ultrafine particles that are produced by grinding and that cling to larger grains dissolve; this is followed by dissolution of the larger grains at sites of excess surface energy. Lagache (1976) argued that in her closed system experiments, the decreasing rate of dissolution is the result of increasing concentra- tions of species in solution.

Clearly, no real consensus exists for the mechanism and kinetics of experimental dissolution of feldspars and other silicates. In addition, natural weathering conditions are nearly impossible to duplicate in the laboratory, and considerable uncertainty attends extrapolation of laboratory results to natural rates and mechanisms of weathering (Petrovic et al., 1976; Holdren et al., 1977; Bemer, 1978). The lack of evidence for a residual layer or precipitate in some studies could possibly be due to certain experimental procedures. Mineral grains in many experiments are subjected to high rates of water flow, agitation, rinsing, or ultrasonic cleaning. These procedures may disturb or remove possible residual layers or precipitates on mineral grains, and may introduce effects not duplicated in the field. Regardless of the difficulties in reproducing natural weathering environments, however, the laboratory experiments indicate two

1.2 .

TIME, YEARS

TIME, YEARS

FIG. 4. Weathering of limestone tombstones with time. (A) Data from Cann (1974). (B) Data for four different types of limestone, from Goodchild (1890).

general observations: (1) feldspars and other silicates dissolve incongruently, leaving some type of residue, and (2) the rate of dissolution decreases with time.

Weathering of man-made structures. Ob- servation of weathering of man-made structures offers an opportunity to study actual weathering processes over a some- what longer time than that observed in experimental studies. Winkler (1966) con- cluded that surface reduction of carbonate stones used in buildings and tombstones of various ages occurred at a constant rate. Goodchild (1890) and Cann (1974) also demonstrated that solution of limestone is a linear function of time (Fig. 4). Apparently, weathering of limestone, which is a congruent dissolution process, proceeds at a constant rate, which varies with climate and the solubility of the limestone.

Matthias’ (1967) study of the weathering of arkose tombstones suggests a decreasing rate of weathering (surface reduction) with


time until about A.D. 1900 (Fig. 5). During this century the rate of weathering of the tombstones has accelerated, possibly because of climatic change or an increase in atmospheric carbon dioxide as suggested by Matthias (1967), or because of changes in acid pollutants or other man-related af- fects.

Weathering of geologic samples. Modem studies of rates of weathering have bene- fited greatly from the rapidly expanding use of radiometric methods to obtain numerical ages and from modern concepts of Quater- nary stratigraphy and history. The ages of rocks and deposits in many areas are still a matter of some guesswork because of the lack of numerical dates, but even in such areas, undated deposits can sometimes be correlated with some confidence to dated deposits in other areas by a variety of relative-age methods. In addition, modem concepts of Quatemary history and climatic change, such as the framework provided by the marine oxygen-isotope record, supply constraints on the ages of many deposits. As a result, weathering rates are commonly constrained by at least some age information, albeit of varying degrees of certainty.

Weathering trends based on chemical analyses have been examined in studies of dated volcanic ash (Hay, 1960; Hay and Jones, 1972; Ruxton, 1968), in studies of weathering rinds on dated volcanic rocks

1 50 100 150

TIME, YEARS

FIG. 5. Weathering of arkosic tombstones with time. Data (mean values) from Matthias (1967). Origin is at approximately 1%5.

(Jackson and Keller, 1970), and in dated deposits containing volcanic rocks (Col- man, 1977, 1981). These studies suggest that the rate of loss of ions (expressed in chemical analyses as oxides) such as SiOZ, CaO, MgO, Na,O, and K,O tends to decrease with time. However, the tendency is commonly inconsistent, and factors other than time appear to have a strong influence on the release and leaching of these ions. The behavior of A1203, Fe,O,, and TiO, during weathering is more difficult to de- tine, because these elements are nearly stable in most weathering environments and because they are commonly incorporated in large measure into secondary products. Fe0 is oxidized to Fe,O, at rates which appear to decrease with time.

In another type of chemical study, Rux- ton (1968) examined the weathering of radiocarbon-dated layers of volcanic ash in Papua, New Guinea. By assuming the A&O3 content to be constant during weathering, he was able to calculate weight losses during weathering from chemical analyses of the fresh and weathered ash. Because crystals in the mineral fraction of the ash beds were not etched during weathering, he assumed that chemical loss due to weathering was entirely from the glass fraction. He calculated the loss of glass by dissolution as a function of time (Fig. 6) according to the following equation:

where C = concentration of glass at time t, C, = original glass concentration, k = a constant, and e = the base of natural logarithms. Ruxton chose an exponential function to express his data (Fig. 6), but logarithmic functions are more commonly used in empirical weathering studies and so are more suitable for comparisons to other studies discussed here. The equation used in Figure 6 for Ruxton’s data is

C - = a + b log(l + t), CO

where a and b are constants, and other variables as before.


TIME, X 1 OJYRS FIG. 6. Weathering of glass in volcanic ashes from

three localities in New Guinea. Data from Ruxton (1968). Ruxton’s curves (of the form e4) are shown as dashed lines; curves of the form a + b log(1 + t) are shown as solid lines for comparison.

Physical measures of weathering are less complex than chemical measures, and in some studies (for example, Colman, 1977), physical measures yield more consistent results. One of the most common physical measurements applied to the study of weathering rates is the thickness of the altered part of a rock, mineral grain, or glass surface.

age used by Hay and Jones accentuates the decrease in weathering rate. Trends in chemical analyses through time for the weathered ash layers were inconsistent, mostly because of large variations in climate among the different sampling sites (Hay and Jones, 1972).

Volcanic glass and ash are nearly ideal for studying the relation between time and physical measures of weathering because alteration of volcanic glass can be conve- niently measured, and volcanic ash can be dated by independent methods. Hay and Jones (1972) studied the weathering of three ash beds on the island of Hawaii. The thickness of palagonite formed by weathering of the outside surfaces and vesicle walls of pumice fragments showed a sys- tematic increase with time (Fig. 7). The rate of increase in thickness is clearly not constant, but decreases drastically with time. The ages used in Figure 7 have been obtained since the Hay and Jones (1972) study (P. W. Lipman, written communication, 1978) and the new age of more than 30,000 years for the oldest ash (the Pahala Ash) COmDared to the 10.000-17.000 minimum

In another study of the weathering of volcanic glass in Hawaii, Moore (1966) examined the rate of palagonization of glassy surfaces of submarine pillow basalts. Ages of the samples were estimated from the ages of the source volcanoes, and from the thickness of the manganese-oxide coating on the samples, which accumulates on stable surfaces at a rate of 3 mm/my (Bender et al., 1966). Moore found that the thickness of the palagonite rind could be expressed by the equation

d = kP,

where d = thickness, t = time, and k = a constant, with bracketing values of 480- 2000 prn/ka for the samples examined (Fig. 8). As shown by Figure 8, the rate of palagonization decreases with time.

Hydration of obsidian is another alteration of volcanic glass that can be physi- cally measured. The thickness of the hydrated layer at an obsidian surface is related

7-m~ .~ to time by the following equation:

0 10 20 30

TIME X 1O’YRS

FIG. 7. Thickness of palagonite for three ashes of different ages on the island of Hawaii. Weathering data from Hay and Jones (1972); age data from P. W. Lip- man (written commun., 1978). Bars represent ranges of measurements given. (A) Outer surfaces of pumice. (B) Vesicle walls.

ROCK-WEATHERING RATES

0 50 100 150 200 250 300

TIME X IO’YRS FIG. 8. Thickness of palagonite as a function of time

for submarine basalt flows. Data from Moore (1966). The two curves represent the limits for the constant proposed by Moore for his equation d = ktln.

d = kt’12,

where d = thickness, t = time, and k = a constant, which depends primarily on temperature during weathering and on the composition of the obsidian (Friedman and Smith, 1960). Obsidian hydration has been used to date archeological artifacts (Fried- man and Smith, 1960), volcanic rocks (Friedman, 1968), and glacial deposits that incorporate obsidian (Pierce et al., 1976). The rate of change of the thickness of the hydrated layer in obsidian decreases with time (Fig. 9).

Data available on the relation between time and the weathering of mineral aggre- gates in rocks are mostly derived from studies of thicknesses of altered zones. A few studies have attempted to document the relation between weathering-rind thickness and time, using a variety of rock types and time control of varying accuracy.

(?emohouz and $01~ (1966) were able to fit a logarithmic curve to rind-thickness data for basalts in Bohemia (Fig. 10). This curve is of the form

d = a log (1 + bt),

CBA

257

A 0 2 4 6 a IO’YR B 0 40 80 120 160 IO’YR C

TIME

FIG. 9. Examples of rates of obsidian hydration. (A) 0, Experimental hydration of obsidian at 100” C. Data from Friedman and others (1966). (B) n , Hydration of obsidian from rhyoiite flows near Mono Craters, California. Data from Friedman (1968). Cl, dated flow used for calibration. (C) Hydration of obsidian near West Yellowstone, Montana. Data from Pierce and others (1976). A, dated obsidian flows used for calibration; A, obsidian-bearing glacial deposits.

where d = rind thickness, t = time, and a and b are constants. The ages for this curve appear to be from estimated ages of glacial deposits and from numerical dates by un- stated methods. The curve clearly indicates a decreasing rate of rind formation through time.

Chinn (1981) was able to document the rate of weathering-rind development on

TIME, X 10’ YEARS R.P. (A) 1 2 3 4 5 6 7 8 9 10

G

4 2.4 6

< vl 2.0 em

3 2,

&, I.6 4w -z Lyj1.2

;ij

gi y’, 0.0

I ! ;

+- z-

4 0.4 IP

P 0 0 lo 40 60 10 loo 120 140 Ma 180 200 220

TIME, X 10’ YEARS I).P. (6 AND C)

FIG. 10. Weathering-rind thickness versus time curves. (A) New Zealand (A), d = .004FL: data from Chinn (1981). (B) Bohemia (0). d = 4.64 log (1 + O.Olf); data from cernohouz and $01~ (1966). (C) West Yellowstone, Montana (m). d = log (0.73 + 0.038r): from Colman and Pierce (1980). Three points on the Bohemia curve beyond 200,000 years have been omitted. Bars represent error limits given in the original references.


sandstone boulders in New Zealand by using radiocarbon-dated deposits for time control (Fig. 10). His results indicate that the rate of rind development slows exponentially according to the equation

t = 1030 d’.24 9

where d = rind thickness in mm, and t = time in years. This is equivalent to

d = 0.004 t0.81.

Several workers have examined the thickness of weathering rinds on rocks in glacial deposits less than about 10,000 years old, using ages estimated from lichen growth curves and other methods. Birke- land (1973) suggested that the relation between rind thickness and time might be linear for rock glacier deposits on Mt. Sop- t-is, Colorado, but the ages of the two deposits considered (early Neoglacial and Pinedale) are uncertain. Carroll (1974) expressed the relation between weathering- rind thickness and time by two straight lines for three ages (estimated from lichen growth curves) of granitic deposits studied in a cirque in the Colorado Front Range. His results indicate a decreasing rate of rind development with time. Carrara and An- drews (1975) obtained ages consistent with lichenometric estimates for deposits containing andesitic rocks in the San Juan Mountains in Colorado by using a square- root time function for weathering rinds:

d = a + bC2,

where t = time, d = rind thickness, and a and b are constants. These data are tenta- tive because of large uncertainties in deposit ages, but they suggest decreasing rates of rock weathering with time.

about the relation between rock weathering and time (Colman, 1977; Colman and Pierce, 1980). These data suggest nonlinear rates of weathering for basalts and andesites. Different types of time control each indicate nonlinear rates of weathering, and can be considered on three levels of certainty: (1) The glacial deposits in one of the areas studied, near West Yellowstone, Montana, have been dated by combined K-Ar and obsidian hydration methods (Pierce et al., 1976). (2) Deposits in some sampling areas, although not dated by numerical methods, can be reasonably correlated with glacial deposits of known age elsewhere. (3) Limits can be placed on age differences between undated deposits within local sequences.

Rind-thickness data for dated deposits near West Yellowstone show that the rate of rind development decreases with time (Fig. 10). A logarithmic curve, similar to that used by Cernohouz and Sole (1966), appears to provide the best fit to the data. Details of the curve construction are discussed by Colman and Pierce (1980). Re- gardless of the actual form of the curve, the relation between rind thickness and time clearly is not linear.

Although numerical time control in areas (other than West Yellowstone) in which rinds were measured is poor, probable ages of deposits can be estimated by correlation with dated deposits elsewhere. Curves that express development rates for weathering rinds were constructed for each of the study areas other than West Yellowstone by using the West Yellowstone curve as a model, and by using the inferred age of one deposit in each area as a calibration point. Deposits inferred to be about 140,000 years old

WEATHERING RINDS ON BASALTS (marine oxygen-isotope stage 6) were used

AND ANDESITES IN THE for calibration. Each curve was constructed

WESTERN UNITED STATES by multiplying the West Yellowstone curve by a constant so that the new curve uassed

- Recent studies of the development of through the calibration point for that area weathering rinds on basalt and andesite (Fig. 11). The assumptions used in this clasts in glacial deposits in the Western method and the construction of the curves United States offer additional information are discussed in more detail by Colman and


I I I I I , , , , , , , , , ,

2.0 - I I , , 1 , , , , , , , ,

1.6 -

Rain&r_ - - - - 3 -- /-

/- -/o- Larsen .

I I _--- _---

-- -- ---- I .A- m_c--- T~zL -I

iii E 0.8-

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z O- - _ 2.0

z

: 5

1.6-

z I- 1.2-

ii? - AO- nt u.o-

”

50 100 150 200 250

TIME X lOa YEARS

FIG. 11. Weathering-rind thickness as a function of time for seven areas in the western United States. The curve for West Yellowstone appears as in Figure 10; the other curves are based on the form of the West Yellowstone curve and the inferred age of one deposit in each area for calibration (0). The position of other deposits (0) determined by rind thickness.

Pierce (1980). Differences in rates of weathering-rind development for different areas suggested by these curves are probably due to differences in climate and rock type. The nonlinearity of the curves is sup- ported by the fact that the ages derived for the glacial deposits not used for calibration are consistent with many paleoclimatic records.

Even without numerical dating or correlation to dated deposits, estimates of age differences between deposits in a sequence provide time control for weathering rates. For example, deposits of the last and of the next-to-last major glacial maxima can be identified in many areas. The marine oxygen-isotope record (Shackleton and Opdyke, 1973) suggests that these glacial

maxima differ in age by a factor of at least 5 (between stages 2 and 4), and perhaps as much as 10 (between stages 2 and 6). If weathering-rind development were a linear function of time, the ratio of rind thicknesses for two deposits in a sequence should be equal to the ratio of their ages.

In the Yakima Valley, Washington, the Domerie Member of the Lakedale Drift represents the last major glacial advance, probably about 14,000 years ago (Porter, 1976). The Indian John Member of the Kit- titas Drift in the same valley appears to be older than the last interglaciation (Porter, 1976; Waitt, 1979; Colman and Pierce, 1980). Weathering rinds in Domerie deposits have an average thickness of 0.25 mm, and rinds in Indian John deposits average


1.05 mm in thickness (Porter, 1976; Colman and Pierce, 1980). The ratio of the rind thicknesses for these two deposits is about 4, whereas the ratio of their ages is probably about ten. Consequently, the rate of rind development in these deposits appears to decrease with time.

Similar arguments apply to the area near McCall, Idaho, where rinds on basaltic stones in Pinedale Till average 0.35 mm, and those in Bull Lake Till average 1.57 mm in thickness (Colman and Pierce, 1980). The Pinedale Till represents the last major glacial maximum in the area, and is probably about 15,000 years old. The Bull Lake Till near McCall has soils and morphology that suggest it is at least as old as the West Yel- lowstone Bull Lake, dated at about 140,000 years old (Pierce et al., 1976). The ratio of the rind thicknesses of the Pinedale and Bull Lake at McCall is about 4.5, whereas the ratio of their ages is probably 10 or more. These interpretations also suggest that the rate of rind development decreases with time.

In summary, weathering-rind studies have provided important control on rates of weathering by using a variety of time estimates. Whether one considers numerical dates, probable ages derived from correla- tions, or possible age differences within local sequences, all available data suggest that rates of rock weathering decrease through time.

MATHEMATICAL ANALOGS FOR ROCK-WEATHERING RATES

The studies previously discussed have documented that most rock-weathering rates decrease through time, and have suggested a variety of empirical expressions for this decrease. The explanation for decreasing rates appears to be closely related to the inhibiting role of residues in weathering processes. The amount of residue is commonly used as a measure of weathering, and the rate of production of the residue is amenable to mathematical

expression, independently of the actual weathering processes.

The nature and origin of weathering residues is presently the subject of considerable debate, particularly in relation to evidence from experimental studies of feldspar dissolution (see previous discussion). However, the common observations of incongruent reactions with decreasing rates virtually require the existence of some sort of residue. The term “residue” is used here in a general sense, without regard to composition or origin. A wide variety of weathering features, ranging in scale from mi- crons (“residual layers” in laboratory weathering experiments) to centimeters (weathering rinds) or larger, can be considered residues. In weathered rocks, indi- vidual minerals are commonly observed to weather inwards from altered zones along edges and cracks (Colman, 1977, 1981).

Access of water to unweathered material is a crucial factor in determining rates of weathering. Weathering residues may slow the movement of water to unweathered material and inhibit diffusion of ions between this material and weathering solutions. If so, the development of a residual layer decreases the rate of weathering (and the rate of production of the residual layer) as the increasing thickness of the layer in- hibits water movement and ionic diffusion.

Because the products or residues of weathering tend to be chemically stable or metastable in the weathering environment, the rate of chemical destruction of such residues is generally small. Physical removal of residues of weathering, on the other hand, may be an important factor affecting the thickness of the residual layer. In particular, this may be true in the case of large-scale (> 10-l m) residues, such as sap- rolite, or in experiments in which samples are mechanically agitated. However, in many weathering environments, especially those below the ground surface, physical disturbances are considerably less than those affecting large-scale surface features and some weathering experiments. For


weathering rinds on basaltic and andesitic stones in soils, physical removal of the rinds (residues) is negligible (Colman, 1977; Colman and Pierce, 1980). If physical removal of residues is minor, the rate of weathering decreases to rates comparable to the destruction of the residue only when sufficient residue has accumulated to se- verely impede the alteration of the fresh material.

Because residues appear to inhibit weathering, the rate of formation of the residual layer at the unweathered mineral surface is inversely proportional to some function of the thickness of the residual layer:

drf 1 -=- dt f(r) ’

(1)

where dr,/dt = the rate of formation of the residual layer of thickness r, andf(r) is any positive function of r. For f(r) of the form km

(2)

where k is a constant, and n is an unspecified exponent greater than zero (r + 1 must be used rather than r for the boundary condition r = 0). If dr,/dt is the rate of destruction of the residual layer at its outer surface, then

$ = k(r + 1))” -2 , (3)

where drldt = the rate of change in the thickness of the residue.

The rate of destruction of the residual

An equilibrium thickness of the residual layer (r,) is reached when drldt = 0, or, from Equation (5) when

r = re = (k/D)“” - 1. (6)

As in most equilibrium calculations, r approaches (r,) asymptotically.

If the rate of destruction of the residual layer (drdldt or D) at the outer surface is very small, long periods of time may be necessary for the residual layer to reach an equilibrium thickness. As D approaches zero, Equation (5) reduces to

dr drf -=-= dt dt

k(r + I)-“. (7)

Integrating, with the boundary condition t = 0, r = 0, yields

kt

( 1

l/t~l+ll r= ~

n+l (8)

which is of the form

r = kt”, (9)

which is a nonlinear function for all m not equal to zero. Because m = ll(n + 1) and 0 < m < 1, the rate of change of r decreases with time.

To fulfill the premise that the rate of formation of the residual layer is inversely proportional to its thickness, the function f(r) in Equation (1) can be any positive function of r. For example, if

f(r) = ke’ (10)

then by the same analysis

r = k In(t), (11)

layer at its outer surface depends on the and if chemical stability of the layer in the weathering environment, not on the thickness of

f(r) = kr

the residual layer, and therefore can be then by the same analysis considered a constant here:

(12)

r = kt’/‘.

Thus,

d’-d -zzz

dt D.

dr z = k(r + l)-” -D.

(13)

(4) Equations (9), (ll), and (13) all represent rates of weathering that decrease through time.

Most of the studies concerning residual (5) products of weathering reviewed earlier


suggest that considerable lengths of time are necessary to reach an equilibrium thickness of the residual layer. For weathering rinds on basaltic and andesitic stones in temperate climates, this period of time is more than 0.5 my, and at some sites may be more than 1.0 my (Colman and Pierce, 1980). Therefore, for considerable lengths of time, the rate of formation of the residual layer is far in excess of the rate of its destruction, and equations such as (9), (ll), and (13) provide satisfactory expressions for the relation between weathering and time.

SUMMARY AND CONCLUSIONS

Weathering features commonly have been assumed to be linear functions of time, either because linearity was the simplest assumption in the absence of time calibration, or because only average rates could be calculated. However, theoretical and empirical evidence fail to support a linear time function for most weathering processes. The data suggest that, of all possible time relations, linear functions should be considered special cases. Although some weathering processes, such as the congruent dissolution of limestone, may be linear functions of time, such processes appear to be exceptions rather than the rule.

Few studies of weathering rates have quantitatively measured both time and rock-weathering features. The few studies that adequately document weathering through time clearly indicate that the rate of change of most rock-weathering features decreases with time. Recent studies of weathering rinds on volcanic clasts in Quaternary deposits in the western United States (Colman and Pierce, 1980) strongly support this conclusion.

Some alteration processes, such as hydration of volcanic glass, acting in isolation from other processes, appear to be best expressed as exponential functions of time, with an exponent of about 0.5. Theoreti- cally, such functions are appropriate for the kinetics of diffusion processes. However,

weathering is generally the result of several interacting processes, for which theoretical models are usually intractable. Empirical expressions of weathering rates commonly provide logarithmic functions of time.

The formation of stable residues during weathering appears to be a major reason that the rate of change of most rock- weathering features decreases with time. Stable weathering products probably impede the flow of water to the unaltered material and slow the transport of elements away from it. As a result, the rate of weathering decreases as the residue is produced.

The residual products of weathering may themselves reach physical equilibrium. An equilibrium thickness of the residual layer is reached when its rate of formation, which decreases as the layer thickens, equals its rate of destruction. However, for small- scale weathering features, or for weathering features below the ground surface, the rate of destruction of the residual layer is commonly small or negligible, and the thickness of the residual layer can be expressed as a simple decreasing function of time. These mathematical analogs of weathering rates represent the conclusion that the more a material weathers, the slower it weathers, and the thicker the residual layer, the slower its thickness increases.

These conclusions have important impli- cations for the use of weathering features as age indicators. Although definition of functional relations between weathering and time is in its infancy, future studies using reliable time calibration offer the potential for the use of weathering features as ap- proximate dating methods. The fact that the rates of most weathering processes decrease with time is in itself useful in es- timating ages of weathered materials, because a constant-rate assumption then provides a limiting age.

ACKNOWLEDGMENTS I especially thank K. L. Pierce for collaboration on

the weathering-rind studies and for discussion of many of the ideas presented in this paper. R. M. Burke, D. P. Dethier, I. Friedman, M. J. Pavich, F. Pessl, Jr.,


and C. A. Wallace provided helpful reviews of earlier versions of the manuscript.

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Rock-weathering rates as functions of time