# 2.31. Carbon Isotopes¶

CLM includes a fully prognostic representation of the fluxes, storage, and isotopic discrimination of the carbon isotopes ^{13}C and ^{14}C. The implementation of the C isotopes capability takes advantage of the CLM hierarchical data structures, replicating the carbon state and flux variable structures at the column and PFT level to track total carbon and both C isotopes separately (see description of data structure hierarchy in Chapter 2). For the most part, fluxes and associated updates to carbon state variables for ^{13}C are calculated directly from the corresponding total C fluxes. Separate calculations are required in a few special cases, such as where isotopic discrimination occurs, or where the necessary isotopic ratios are undefined. The general approach for ^{13}C flux and state variable calculation is described here, followed by a description of all the places where special calculations are required.

## 2.31.1. General Form for Calculating ^{13}C and ^{14}C Flux¶

In general, the flux of ^{13}C and corresponding to a given flux of total C (\({CF}_{13C}\) and \({CF}_{totC}\), respectively) is determined by \({CF}_{totC}\), the masses of ^{13}C and total C in the upstream pools (\({CS}_{13C\_up}\) and \({CS}_{totC\_up}\), respectively, i.e. the pools *from which* the fluxes of ^{13}C and total C originate), and a fractionation factor, \({f}_{frac}\):

If the \({f}_{frac}\) = 1.0 (no fractionation), then the fluxes \({CF}_{13C}\) and \({CF}_{totC}\) will be in simple proportion to the masses \({CS}_{13C\_up}\) and \({CS}_{totC\_up}\). Values of \({f}_{frac} < 1.0\) indicate a discrimination against the heavier isotope (^{13}C) in the flux-generating process, while \({f}_{frac}\) \(>\) 1.0 would indicate a preference for the heavier isotope. Currently, in all cases where Eq. is used to calculate a ^{13}C flux, \({f}_{frac}\) is set to 1.0.

For ^{14}C, no fractionation is used in either the initial photosynthetic step, nor in subsequent fluxes from upstream to downstream pools; as discussed below, this is because observations of ^{14} C are typically described in units that implicitly correct out the fractionation of ^{14}C by referencing them to ^{13}C ratios.

## 2.31.2. Isotope Symbols, Units, and Reference Standards¶

Carbon has two primary stable isotopes, ^{12}C and ^{13}C. ^{12}C is the most abundant, comprising about 99% of all carbon. The isotope ratio of a compound, \({R}_{A}\), is the mass ratio of the rare isotope to the abundant isotope

Carbon isotope ratios are often expressed using delta notation, \(\delta\). The \(\delta^{13}\)C value of a compound A, \(\delta^{13}\)C_{A}, is the difference between the isotope ratio of the compound, \({R}_{A}\), and that of the Pee Dee Belemnite standard, \({R}_{PDB}\), in parts per thousand

where \({R}_{PDB}\) = 0.0112372, and units of \(\delta\) are per mil (‰).

Isotopic fractionation can be expressed in several ways. One expression of the fractionation factor is with alpha (\(\alpha\)) notation. For example, the equilibrium fractionation between two reservoirs A and B can be written as:

This can also be expressed using epsilon notation (\(\epsilon\)), where

In other words, if \({\epsilon }_{A-B} = 4.4\) ‰ , then \({\alpha}_{A-B} =1.0044\).

In addition to the stable isotopes ^{1}^{2}C and ^{13}C, the unstable isotope ^{14}C is included in CLM. ^{14}C can also be described using the delta notation:

However, observations of ^{14}C are typically fractionation-corrected using the following notation:

where \(\delta^{14}\)C is the measured isotopic fraction and \(\mathrm{\Delta}^{14}\)C corrects for mass-dependent isotopic fractionation processes (assumed to be 0.975 for fractionation of ^{13}C by photosynthesis). CLM assumes a background preindustrial atmospheric ^{14}C /C ratio of 10^{-12}, which is used for A:sub::abs. For the reference standard A\({}_{abs}\), which is a plant tissue and has a \(\delta^{13}\)C value is \(\mathrm{-}\)25 ‰ due to photosynthetic discrimination, \(\delta\)^{14}C = \(\mathrm{\Delta}\)^{14}C. For CLM, in order to use the ^{14}C model independently of the ^{13}C model, for the ^{14}C calculations, this fractionation is set to zero, such that the 0.975 term becomes 1, the \(\delta^{13}\)C term (for the calculation of \(\delta^{14}\)C only) becomes 0, and thus \(\delta^{14}\)C = \(\mathrm{\Delta}\)^{14}C.

## 2.31.3. Carbon Isotope Discrimination During Photosynthesis¶

Photosynthesis is modeled in CLM as a two-step process: diffusion of CO_{2} into the stomatal cavity, followed by enzymatic fixation (Chapter 2.9). Each step is associated with a kinetic isotope effect. The kinetic isotope effect during diffusion of CO_{2} through the stomatal opening is 4.4‰. The kinetic isotope effect during fixation of CO_{2} with Rubisco is \(\sim\)30‰; however, since about 5-10% of carbon in C3 plants reacts with phosphoenolpyruvate carboxylase (PEPC) (Melzer and O’Leary, 1987), the net kinetic isotope effect during fixation is \(\sim\)27‰ for C3 plants. In C4 plant photosynthesis, only the diffusion effect is important. The fractionation factor equations for C3 and C4 plants are given below:

For C4 plants,

For C3 plants,

where \({\alpha }_{psn}\) is the fractionation factor, and \(c^*_i\) and pCO_{2} are the revised intracellular and atmospheric CO_{2} partial pressure, respectively.

As can be seen from the above equation, kinetic isotope effect during fixation of CO_{2} is dependent on the intracellular CO_{2} concentration, which in turn depends on the net carbon assimilation. That is calculated during the photosynthesis calculation as follows:

where \(a_n\) is net carbon assimilation during photosynthesis, \(p\) is atmospheric pressure, \(g_b\) is leaf boundary layer conductance, and \(g_s\) is leaf stomatal conductance.

Isotopic fractionation code is compatible with multi-layered canopy parameterization; i.e., it is possible to calculate varying discrimination rates for each layer of a multi-layered canopy. However, as with the rest of the photosynthesis model, the number of canopy layers is currently set to one by default.

## 2.31.4. ^{14}C radioactive decay and historical atmospheric ^{14}C and ^{13}C concentrations¶

In the preindustrial biosphere, radioactive decay of ^{14}C in carbon pools allows dating of long-term age since photosynthetic uptake; while over the 20\({}^{th}\) century, radiocarbon in the atmosphere was first diluted by radiocarbon-free fossil fuels and then enriched by aboveground thermonuclear testing to approximately double its long-term mean concentration. CLM includes both of these processes to allow comparison of carbon that may vary on multiple timescales with observed values.

For radioactive decay, at each timestep all ^{14}C pools are reduced at a rate of –log/\(\tau\), where \(\tau\) is the half-life (Libby half-life value of 5568 years). In order to rapidly equilibrate the long-lived pools during accelerated decomposition spinup, the radioactive decay of the accelerated pools is also accelerated by the same degree as the decomposition, such that the ^{14}C value of these pools is in equilibrium when taken out of the spinup mode.

For variation of atmospheric ^{14}C and ^{13}C over the historical period, \(\mathrm{\Delta}\)^{14}C and \(\mathrm{\Delta}\)^{13}C values can be set to either fixed concentrations or time-varying concentrations read in from a file. A default file is provided that spans the historical period (Graven et al., 2017). For \(\mathrm{\Delta}\)^{14}C, values are provided and read in for three latitude bands (30°N–90°N, 30°S–30°N, and 30°S–90°S).