Deriving the flux equation: the model

At constant pressure, the total rate at which water evaporates into the chamber sf_w (mol s^-1) is balanced by a small flow rate of air out of the chamber u (mol s^-1). The CO₂ mole fraction of the air outside the chamber is c_a, inside the chamber is c_c, and in the soil is c_s, all in mol mol^-1. The chamber air water vapor mole fraction is w_c (mol mol^-1). The rate constant k (s^-1) characterizes leaks (if any) due to diffusion of CO₂ between the soil chamber and outside air. The chamber volume v includes the volume of the pump and measurement loop.

The mass balance equations for CO₂, water vapor, and air take the form

1‑4storage = flux in - flux out

We neglect the effects of leaks for now, but we will consider them later.

CO₂ mass balance

1‑5

H₂O mass balance

1‑6

Air mass balance

1‑7

is the number density of CO₂ in the chamber, is the number density of water vapor in the chamber, and is the total number density of air in the chamber (all in mol m^-3); , where is the number density of dry air in the chamber.

The number density of air is given by the ideal gas law, , where R is the gas constant (8.314 Pa m³ K^-1 mol^-1), and T_K is the absolute temperature (K). From equation 1‑7, with ρ and T_K constant, and sf_w >> sf_c,

1‑8

Combining equations 1‑6 and 1‑8, and noting that , we find

1‑9

Combining equations 1‑5, 1‑8 and 1‑9 gives

1‑10

Equation 1‑10 has the same form as that used in the LI‑6400 Portable Photosynthesis System for soil respiration; however, it can be simplified by defining , which is the chamber CO₂ mole fraction corrected for water vapor dilution. This is called C_dry (ppm) in the LI-8100A data output.

Differentiating we find,

1‑11

Substituting this into equation 1‑10 gives

1‑12

Equation 1‑12 has an important advantage over equation 1‑10 because it is not necessary to estimate the rate of increase in water vapor mole fraction. In most measurements, the water vapor mole fraction increases in a highly non-linear fashion, and the rate is estimated with a linear function. Thus, in effect, equation 1‑10 forces us to use average values for and . But with equation 1‑12, the dilution correction is made point-by-point, and estimates of the initial values at time zero are used to estimate f_c at the instant the chamber closed. This is both easier and more accurate than the procedure required to implement equation 1‑10.

In order to use equation 1‑12 the initial values must be known for ρ and T_K (to compute ρ_c), as well as the initial values for w_c and . After the chamber closes, the LI-8100A performs a linear regression with time on the first 10 values of each measured variable. The initial values of ρ, T_K and w_c are obtained from the time zero intercepts of these regressions; however, finding the initial value for requires a little more work.

To do this, f_c is defined in terms of the CO₂ mole fraction gradient across the soil- to-chamber interface and a transfer coefficient, to obtain

1‑13

where c_s is the CO₂ mole fraction in the soil surface layer communicating with the chamber (mol mol^-1), g is conductance to CO₂ (m s^-1), and ρ_c is the density of air (mol m^-3). The soil and chamber must be isothermal for equation 1‑13 to hold.

Combining equation 1‑13 with equation 1‑12, considering all variables except c_c' to be constant, and rearranging, gives

1‑14

where . When w_c = w_s , c_s', gives the dilution-corrected CO₂ mole fraction in the soil layer communicating with the chamber. We do not expect w_c to equal w_s exactly, but most of the time they will differ by less than 0.02 mol mol^-1 or so, which introduces only a small uncertainty in c_s'. If c_s' is taken as a constant, then equation 1‑14 can be integrated to give

1‑15

where A = sg v^-1 is a rate constant (s^-1) and c_c'(0) is the initial value of the dilution- corrected CO₂ mole fraction when the chamber closes. The rate of change in c_c'(t) at any time can be computed from the derivative of equation 1‑15.

1‑16

Calculating the flux from measured data

In the LI-8100A, equations 1‑12, 1‑14 and 1‑15 are implemented in a form that presents the variables in more familiar and intuitive units. Equation 1‑12 is computed as

1‑17

where F_c is the soil CO₂ efflux rate (μmol m^-2 s^-1), V is volume (cm³), P₀ is the initial pressure (kPa), W₀ is the initial water vapor mole fraction (mmol mol^-1), S is soil surface area (cm²), T₀ is initial air temperature (°C), and is the initial rate of change in water-corrected CO₂ mole fraction (μmol mol^-1).

Examine Figure 1‑3 to see C'(t) vs t data that were obtained from a soil CO₂ flux measurement with two observations. The data are marked to show when the chamber closed and when it opened.

The dead band is the time until steady chamber mixing is established, and typically lasts 10s to 30s. After mixing is stable, the data are fit with an empirical equation that has a form similar to equation 1‑15:

1‑18

where C'(t) is the instantaneous water-corrected chamber CO₂ mole fraction, C₀' is the value of C'(t) when the chamber closed, and C_x' is a parameter that defines the asymptote, all in μmol CO₂ per mol dry air (µmol mol^-1); a is a parameter that defines the curvature of the fit (s^-1).

The initial value of C'(t), called C₀' in equation 1‑18, is computed from the intercept of a linear regression of the first 10 points after the chamber closes. This is used as a parameter in the non-linear regression that fits equation 1‑12 to the C'(t) vs t data between the end of the Dead Band and the end of the observation. This regression yields values for the parameters C_x', a and t₀. t = t₀ represents the time when C'(t) in equation 1‑18 equals its initial value when the chamber closes, or C'(t₀) = C₀'. The delay between the instant the chamber closed and t₀ gives the time required to establish steady mixing. CO₂ offsets or time delays can occur when the chamber closes, and these events can cause t₀ to be positive or negative in value.

All the initial values needed to obtain the soil CO₂ efflux rate, F_c, in equation 1‑12 can now be computed. The initial values P₀, T₀ and W₀ are all obtained from the intercepts of linear regressions of the first 10 measurements of P, T and W after the chamber closes. The rate of change of dilution-corrected chamber CO₂ mole fraction can be computed at any time from

1‑19

When t = t₀,

1‑20

Equation 1‑20 gives an estimate of the rate of change in C' at the instant the chamber closed. This value must be estimated mathematically. It cannot be measured directly at any time during the measurement because imperfect mixing prevents an accurate estimate early in the measurement cycle, and later in the cycle, the increasing chamber CO₂ concentration continuously reduces the gradient between soil and chamber. This suppresses the rate, as can be seen from equation 1‑19 and also in Figure 1‑3.

Relationship between the model and the empirical equation

The diffusion model provides an equation with a form that allows correction for the effect of changing gradients on the rate, which in turn, makes it possible to estimate the initial rate. It is worthwhile to distinguish between the model function given in equation 1‑15 and the empirical function in equation 1‑18. As just described, the units are different in the two expressions; but more important, for the parameters c_s and A in equation 1‑15 to have their defined meaning, the assumptions underlying the derivation must be true. By contrast, equations 1‑18 through 1‑20 are treated as empirical functions and are used only to estimate the CO₂ rate of change, dC'/dt. The parameters C_x', a and t₀ do not depend upon a specific theoretical interpretation, and may or may not provide reliable estimates of soil parameters.

Correcting for initial CO₂ concentrations that differ between measurements

Different measurements may begin at different CO₂ concentrations, which introduces variation into the data, because the flux rate changes with chamber CO₂ concentration. Correcting the measurements to a common target CO₂ concentration may reduce such variation. For a given curve fit, the CO₂ rate of change can be computed at any CO₂ concentration according to

1‑21

This calculation is supported in the SoilFluxPro software, which is included with the LI-8100A.

Evaluation of other methods for computing soil CO₂ efflux

Other approaches have been used for computing CO₂ flux in transient measurements. One commonly used method is to fit a linear function to what is sometimes referred to as "the linear portion" of the curve. Unfortunately, there is no linear portion, as can be seen from careful inspection of Figure 1‑3. The slope is meaningless in the initial phase before steady mixing is established, and after steady mixing is established, the extent of the non-linearity depends upon the soil surface-to-chamber volume ratio and the flux rate. During this time, CO₂ vs time curves are always concave in a downward direction, meaning that linear regression over this portion of the data set will give an underestimate of the rate of change. In every case we have tested so far, the average rate measured by linear regression is less than the initial rate measured by non-linear regression. Nevertheless, linear regression is a robust numerical approach and the mean values for the CO₂ efflux rate reported by the LI-8100A in Type 3 records are computed by this method. We recommend you use these only for comparison to the initial values, which are obtained by fitting equation 1‑18 to the data using a non-linear regression method.

Another approach that has been used to estimate the initial rate is to fit a polynomial to the CO₂ concentration vs time data. This approach is theoretically sound inasmuch as a power series can be generated from a Taylor series approximation to equation 1‑18. Usually, the data are fit with a quadratic equation. We tested this approach and found that while it can be justified on theoretical grounds, it does not work very well in practice. The shape of even a second order polynomial is sensitive to small perturbations in the data. This makes initial rate calculations subject to much larger variations than when the same data are analyzed by nonlinear regression using equation 1‑18.

Effects of high chamber CO₂ concentrations

Finally, we consider the importance of choosing appropriate observation times and pre-purge times. We do not have experience on all soil types, and cannot give absolute recommendations for the best observation length in all situations.

Nevertheless, our experience so far suggests that 60s to 120s will often work well. This prevents large build-ups in chamber CO₂ concentration at typical change rates such as 0.5 ppm s^-1. We have found that optimal dead band length can vary from about 10s to 60s, with 30s being a good value to use as a first estimate.

Dead bands and observation times can be adjusted after the fact, using SoilFluxPro software (formerly FV8100). This program provides curve fit analysis tools that can be used to find the optimal dead band lengths and observation lengths. Therefore, it is not critical to choose the right dead band and observation length in the field; as long as the observation lengths are long enough, they can be optimized later, if necessary.

Long observations can have the effect of capping the soil and causing the CO₂ concentration to build up in the soil under the chamber. This phenomenon can be observed by performing a sequence of observations in which the chamber concentration is allowed to increase several hundred ppm during each observation. When the pre-purge is set to be just long enough to allow the chamber atmosphere to come back to the ambient CO₂ concentration, the initial rates in sequential observations can often be observed to increase as the soil CO₂ concentration increases. This is expected according to equation 1‑20, dC_c'/dt = a(C_x' - C_ambient'), if it is assumed that C_x' = C_soil'. Thus, long observation lengths may perturb the very process we wish to measure.

Another effect of high chamber CO₂ concentrations is to promote leaks between the chamber and atmosphere. Leaks can be ignored when the gradient between the chamber atmosphere and ambient atmosphere are small. But when the gradient is not small, leaks cannot be neglected and it can be shown that the parameters in equation 1‑15 are altered to become

1‑22

where k and c_a' are the leak rate time constant and water-corrected ambient CO₂ concentration, respectively, and the expression for c_x' replaces c_s' in equation 1‑15. Thus, when chamber CO₂ concentrations are high, the rate constant and asymptote will reflect leaks from the system. Variables that are stored in data files are described in Table 6‑4.