# 2.32. 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.

## 2.32.1. 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 ( and ,
respectively) is determined by , the masses of
^{13}C and total C in the upstream pools
( and ,
respectively, i.e. the pools *from which* the fluxes of
^{13}C and total C originate), and a fractionation factor,
:

(2.32.1)¶

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

For ^{1}^{4}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 ^{1}^{4}C are typically
described in units that implicitly correct out the fractionation of
^{1}^{4}C by referencing them to
^{1}^{3}C ratios.

## 2.32.2. 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,
, is the mass ratio of the rare isotope to the abundant isotope

(2.32.2)¶

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

(2.32.3)¶

where = 0.0112372, and units of are per mil (‰).

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

(2.32.4)¶

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

(2.32.5)¶

In other words, if ‰ , then .

In addition to the stable isotopes ^{1}^{2}C and ^{1}^{3}C, the unstable isotope
^{1}^{4}C is included in CLM. ^{1}^{4}C can also be described using the delta notation:

(2.32.6)¶

However, observations of ^{1}^{4}C are typically
fractionation-corrected using the following notation:

(2.32.7)¶

where C is the measured isotopic
fraction and 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, which is a plant tissue and has
a C value is 25 ‰ due to photosynthetic discrimination,
^{14}C = ^{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
C term (for the calculation of
C only) becomes 0, and thus
C = ^{14}C.

## 2.32.3. 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
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
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,

(2.32.8)¶

For C3 plants,

(2.32.9)¶

where is the fractionation factor, and
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:

(2.32.10)¶

where is net carbon assimilation during photosynthesis, is atmospheric pressure, is leaf boundary layer conductance, and 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.

## 2.32.4. ^{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 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/, where 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, ^{14}C and :sup:13C 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
^{14}C, values are provided and read in for three latitude bands (30 ^{o}N-90 ^{o}N, 30 ^{o}S-30 ^{o}N, and 30 ^{o}S-90 ^{o}S).