Radiogenic Isotopes as Tracers of Sources and Mixing in the Solid Earth, Hydrosphere, Biosphere

Transcription

1 Radiogenic Isotopes as Tracers of Sources and Mixing in the Solid Earth, Hydrosphere, Biosphere Reading: White, Lecture 22, part on the mathematics of binary mixing. Also: Faure, 1986, Chs. 9, 11 OR... Faure and Mensing, 2005, Chapter 16 and Chapter 19 Motivation: We have already seen that different reservoirs on earth have different radiogenic isotope contents. For example, old crustal rocks with much Rb have large 87 Sr/ 86 Sr ratios (e.g., >0.712), whereas magmas recently ascended from the mantle have low 87 Sr/ 86 Sr ratios (e.g., <0.707). Spend some time recalling how if the parent element to daughter element ratio (e.g., Rb/Sr ratio) is large, the material becomes more radiogenic over time relative to other materials with smaller ratios. Because different parts of the earth have different isotope ratios, we can sometimes learn where the Sr, Nd, Pb, etc. in a rock, water, or biological sample came from. In some cases, there are two potential sources, and we can determine how much each source contributed to the sample we have. Examples: 1) Groundwater sampled at point S may have originated in 2 potential recharge areas, A and B. If these recharge areas impart different isotope ratios to the water, we may be able to determine the source. 2) A bird s feather may contain isotope ratios inherited from a distant feeding ground and thus the isotopes may be used to track migration. 3) Sr in a groundwater system may be derived from weathering of a) silicate rocks and/or b) carbonate rocks. The relative sizes of the two contributions might be estimated by Sr isotope analysis. 4) A magma may be a mixture of a) material derived directly from the mantle and b) crustal contamination. We expect the crustal component to contain Sr that is more radiogenic than the mantle component and this we can determine how much of it is in the mixture. IMPORTANT NOTE: Often, people refer to the distinct isotope ratios as isotopic signatures or isotopic fingerprints. BEWARE! Isotope ratios are not as immutable as signatures or fingerprints, but if we are careful, we can learn about magma sources, solute sources and chemical reactions. Guide Questions: 1. How can we predict whether a given rock will have a highly radiogenic isotope value or a less radiogenic one? 2. If the various rocks or waters of a system are formed by various mixtures of two materials, how do we expect the mixtures compositions to plot on a graph of isotope ratio versus concentration? ersus 1/conc.? 3. If the various rocks or waters of a system are formed by various mixtures of two liquids, how do we expect the mixtures compositions to plot on a plot of isotope ratio of one element versus isotope ratio of a different element? 4. What is the exact meaning of residence time as applied to a well-mixed reservoir? 5. How long does it take for the water in the oceans to become completely mixed?

2 6. What are the important inputs for Sr in the oceans? 7. What geological events seem to influence the 87 Sr/ 86 Sr ratio of the oceans? 8. Picture a reservoir with Sr inputs and outputs, in which the Sr concentration is at steady state. Write a differential equation that expresses the rate of change in the 87 Sr/ 86 Sr ratio of the reservoir in response to a given input of Sr. 9. What assumption do we almost always make for the outputs of radiogenic isotopes from the reservoir? 10. If the 87 Sr/ 86 Sr ratio of one input to the reservoir suddenly changes, then remains constant, how does the 87 Sr/ 86 Sr of the reservoir evolve over time? What is the final value? 11. If groundwater with a given 87 Sr/ 86 Sr ratio flows into and begins reacting with a new rock type, how does the water s 87 Sr/ 86 Sr evolve as the water flows along? What governing equation describes this? Recognizing and modeling mixing of two materials with isotope data Often, when we look at data from solutes, we see that they reflect mixture of multiple sources of solute. So we will need mathematical tools to analyze data that result from mixing. Binary Mixing How can we analyze data from a system that is a mixture of sources? 1) Binary Mixing: Take two vats, mix them completely, look at the mixture. A + B = M The following equations are derived from simple mass balance statements and algebra. See the appendix at the end of these notes for derivations. The concentration of the mixture is given by: = f + (1-f) where f is the volume fraction of A in the mixture (f = A / ) C Pure B Pure A f The equation relating concentrations to isotope ratios is: ª 1 Ê + R - R ˆ B A Á Ë -C B + R C - R A B B

3 This is an approximation; to be exact, the concentrations would be of the denominator isotope only (e.g., 86 Sr). But this approximation is so nearly perfect we use it in almost all cases. The form of this equation is a hyperbola: ª 1 a + b R C So If you think your data array might reflect mixing of two waters (or magmas, etc.), you can test this hypothesis by plotting the results on this type of diagram: R 1/C Also: Plots of one isotope ratio versus another also produce hyperbolic curves for mixtures. 143 Nd/ 144 Nd 187 Sr/ 86 Sr Note that the curvature depends on the relative concentrations of Sr and Nd in the two end members: See the beginning of chapter 22 in White, or section 9.5 in Faure, 1986 for equations.

4 Use of these mixing models (Practical procedure): 1. Measure isotope and/or concentration data for various parts of a system, and plot as shown above. 2. Do the data fit a mixing model? 3. If no, then mixing can be ruled out as the sole process that produced the variation. 4. If yes, then mixing is a plausible model and one may be able to calculate the relative contributions of two sources in each sample. HOWEER, THIS DOES NOT MEAN THAT MIXING IS THE ONLY WAY TO GENERATE THE OBSERED PATTERN. MANY PEOPLE OBSERE THAT THEIR DATA FIT A MIXING MODEL AND THEN DECLARE THAT BINARY MIXING MUST BE THE PROCESS RESPONSIBLE FOR THE ARIATION. YOU SHOULD ONLY SAY THIS IF YOU ARE SURE THAT ALTERNATIE MECHANISMS ARE NOT PLAUSIBLE. Temporal variation of well-mixed reservoirs with varying inputs Example: 87 Sr/ 86 Sr in the Oceans: See variation of 87 Sr/ 86 Sr over time: Faure and Mensing, 2005, Fig Large variations over time to Some of the changes are fairly rapid 3. Long-term trends, too (e.g., steady increase since 160 Ma) What controls the 87 Sr/ 86 Sr as a function of time? Sr/ 86 Sr of various inputs a) Continental weathering- generally moderate to high o Carbonate rocks to o Silicate Rocks old granites, shales, metamorphic rocks have high ratios (e.g., >0.712) younger ones not so large (e.g., to 0.712) b) Mid-ocean ridge hydrothermal systems add Sr to water- low 87 Sr/ 86 Sr (~0.704) c) Dissolution of carbonate buried in ocean sediments may have occurred during some time periods (?) 2. Sizes of these inputs change over time 3. Memory of past 87 Sr/ 86 Sr in the oceans - residence time of Sr is 2-3 Ma 4. Ignore outputs: 87 Sr/ 86 Sr of output = ocean 87 Sr/ 86 Sr; no direct effect on ocean 87 Sr/ 86 Sr (Note this only works when the oceans are well mixed so that the outputs are isotopically identical to the ocean as a whole.) Why is the 87 Sr/ 86 Sr record in the oceans of interest? 1) We think that continental collisions increase weathering of silicate material, and this extra weathering affects the chemistry of the ocean in ways that decreases CO 2 in the atmosphere and thus leads to cooler climate conditions. 2) Age dating of marine carbonate rocks. 87 Sr/ 86 Sr ratios measured in carbonate rocks can be matched to the record to give precise age dates. The dates are often non-unique (e.g., occurs many times in the last 500 million years), but usually we start with a rough idea of age from the rock layer sequences.

5 Mathematical tools for time series data from systems with constant inputs that may vary over time What if we have a system where small amounts of liquid are added in continuously and mixed, instead of an instantaneous mixing of two vats??? (This is more like a natural situation) Develop differential equations for incremental mixing of inputs into a system. Again: Let s start with 87 Sr/ 86 Sr in the oceans: What determines the 87 Sr/ 86 Sr in the oceans at any given time? 1) The ratios of the various inputs NOW. 2) Their sizes NOW. 3) The 87 Sr/ 86 Sr already in the system from past additions. Model the ocean as a well-mixed reservoir. Well-mixed = completely homogenized each instant. Then we need to understand how mixing in of an input affects the reservoir. We want to know either: 1) The steady-state value of the ocean OR, if the ocean is not in a steady state 2) The ratio as a function of time R(t) Here s a model of the oceans. 2 inputs of Sr, one output. Now, we must formulate an equation that expresses how the system responds to inputs and outputs: The question: How does the isotope ratio of a reservoir of substance A change as an increment of substance B is mixed in? Let c = concentration in A, c in = concentration in B, R = isotope ratio in A, R in = isotope ratio in B. is the volume of A, is the incremental volume of B added. I use volume so that this is easy to visualize. Here s the differential equation for changes in an isotope ratio with addition of volume increments of B. See appendix for the derivation. In a strict sense, all the concentrations and masses would be of the

6 denominator isotope (e.g., 86 Sr) only, but using total elemental concentrations is a very close approximation. dt = dt c c in [ R in - R] = added mass 2 ( isot. ratio constrast) mass 2 dt = dm in dt m = dm in m [ R - R], in [ R - R] we can add more terms for multiple inputs = dm i in m  i [ R - R] i Note: This only applies if outputs have the same R as the reservoir. If isotopic fractionation causes the outputs to have different R than the reservoir, this analysis does not apply. However, for all radiogenic isotopes, it works. For cases where outputs do not have the same isotope ratios as the reservoir (e.g., oxygen isotopes or other light stable isotopes), the output must be included as well. For the outputs, dm in <0. Works for delta notation too! dd dt =  i dm i dt m d i -d [ ] What types of systems does this apply to? 1) The ocean, or lakes. 2) Groundwater or pore fluids that are receiving and losing solute as they react with the host rock 3) Magma contaminated by country rock over some time period. Example #1: What is the steady state 87 Sr/ 86 Sr ratio of a reservoir? We start with the governing equation: dm in dt = 0 = dt [ m R - R in ] /dt = 0 because steady state means no change with time. On the left side, the only term that can be zero is the R in -R term. So the system reaches steady state when the reservoir R equals the input (or the weighted average of all inputs if there are multiple inputs). Example #2: Start with a crude model of the ocean: Sr budget 1) Input from weathering 2) Output via carbonates What are the 87 Sr/ 86 Sr ratios of these? Draw the vat model. What do you think will happen to the vat as a function of time, starting with some arbitrary R, if we abruptly change the input to a higher 87 Sr/ 86 Sr value? For simplicity, assume the Sr concentration in the ocean is constant over time and ocean volume is also constant.

7 dt = dt c c in [ R in - R] OR 1 [ R in - R] dt = J c in in c, J is input rate, volume per unit time. Solution: Intuitively, you can guess it is a function that asymptotically approaches R Noting that the change in R is closely related to R itself, you might guess the solution is a negative exponential that asymptotically approaches R. R = ( R 0 - R in )e (- t / t ) + R in, t = c 2, NOTE: t = total mass/mass flux in (= residence time!!!!!) J in c in Definition of Residence time: The time spent in the system by an average atom of the element you are tracking. Calculation of residence time: = total mass in the system divided by the rate of mass input. This assumes the concentrations of the input and the reservoir are constants. In reality, these concentrations would probably change if something caused the isotope ratio of the input to change. How can we model more complex cases? The governing equation can be solved for constantly variable inputs and outputs. We can use numerical integration for the differential equation OR we can use fourier transforms, maybe other techniques as well. Example #3: How does groundwater evolve as it reacts with its host rock and flows along? What do we expect to see in terms of a spatial pattern? Imagine a packet of water moving along. Solute is added to it by dissolution of rock. This can also be modeled as a mixing process. dt = d m dt 1 [ R - R], c 2 dt = J [ R - R], where J is the reaction flux per liter water c 2 Same result as before- exponential decay.disequilibrium decreases by a factor of e over some distance- what is that distance? Time constant is c2/j. Distance would then be u*c2/j. What if input is not uniform? Use numerical solution. Appendix 1: Derivation of binary mixing equations. First: Mixing gives linear relationship between volume of each liquid and concentration: m A + m B = m M (1) (m is mass) A + = (2) (C is concentration, is volume) A / + / = (3)

8 Define f = A / (4) = f + (1-f) (5) This gives us the concentration of the mixture. Then: Isotope Ratios DO NOT mix linearly Write 2 statements of eqn 5: 1 for each isotope:, 1 =, 1 f +, 1 (1-f) (6), 2 =, 2 f +, 2 (1-f) (7) = C A,1 + M,1 B,1 = M (8),2 A,2 + B,2 M =,1 A +,1,2 A +, 2 (9) =,1 A +,1,2 (but because R = C 1 /C 2 ) (10) = R A, 2 A,2,2 (11),2 = R A,2 A,2 (12) A (very good approximation), or m M m A m B (13) now get rid of s, so we can plot R vs. C: A, f (1- f ) but, using equation (5) above, f = - - (1- ) Only variable on right side is. Divide both sides by, collect terms ª 1 R A Ê + R - R ˆ B A Á Ë -C B + R C - R A B B R B - R B R B This gives some terms that are constant and some that are 1/ terms. So the form of the equation is: ª 1 a + b

9 Appendix 2: Derivation of governing equation for the changes of isotope ratio in a reservoir over time: dc dt = 1 dt (c -c), (makes sense intuitively) dc in = 1 c 2 dc 1 = 1 (c in - c) (1) define R = c 1, - c 1 dc 2, (using the product rule) c 2 ( c 2 ) 2 = 1 È dc 1 - R dc Í 2, subst. for c derivatives using (1) c 2 Î Í = 1 È 1 ( c 2 c 1,in -c 1 ) - R 1 c Í ( 2,in - c 2 ), but Rc Î 2 = c 1, also pull out the term = 1 [( c c 1,in -c 1 ) - Rc 2,in + c 1 ], simplify and pull out c 2, = c È 2,in c Í 1,in - R 2 c 2 Î Í c 2,in and = c 2,in R - R c 2 [ ] and to a very close approximation ª c in c [ R - R]