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Soil Science Society of America Journal 67:20-31 (2003)
© 2003 Soil Science Society of America

DIVISION S-1—SOIL PHYSICS

Oxygen Transport to Plant Roots

Modeling for Physical Understanding of Soil Aeration

F. J. Cook*,a and J. H. Knightb

a CSIRO, Land and Water, 120 Meiers Road, Indooroopilly QLD 4068, Australia and CRC for Sustainable Sugar Production
b CSIRO, Land and Water, P.O. Box 1666, Canberra ACT 2601, Australia

* Corresponding author (freeman.cook{at}csiro.au)


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
A model that couples the diffusion of O2 to plant roots at the microscale to diffusion of O2 through the soil at the macroscale is derived. The solution is complex, but if the length scales (Zr and Zm; see APPENDIX for list of symbols) are equated a simple analytical expression can be obtained. This model is used to investigate relationship between a critical air-filled porosity ({theta}m) and the other parameters; viz temperature (T), O2 concentration in the bulk soil (C*), O2 concentration at the root surface (Cr), root length density (L), the ratio of root radius (a) to the water film radius (R), microbial respiration (Mo), and length scales (Zm and Zr) related to the depth to which microbial and plant respiration are active in the model using sensitivity analysis. The model shows that {theta}m is not very sensitive to the O2 concentration at the root surface (Cr), or the ratio of root radius (a)/water film radius (R), but is sensitive to all the other parameters in some part of their range. The results indicate that indices used to define soil aeration; O2 diffusion rate (ODR) or O2 flux, O2 concentration, or air-filled porosity, which have been previously used, are related and a single critical value for these is unlikely. If a constant critical value exists for one of these indexes it cannot exist for the other two. It is also shown that it is highly unlikely that a universal critical parameter related to soil aeration exists for any of these parameters. It is concluded that more parameters than ODR, O2 concentration, or air-filled porosity need to be measured if progress in soil aeration research is to be made.

Abbreviations: NLWR, nonlimiting water range • ODR, O2 diffusion rate


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
THE CONCEPT of a nonlimiting water range (NLWR) was introduced by Letey (1985). This has been recently called the least limiting water range by da Silva and Kay (1997a)(1997b). For the NLWR to be identified the water content at which O2 supply to plants becomes limiting needs to be known or measured. The introduction and attempted uses of the NLWR concept has again highlighted the lack of a good definition for soil aeration. Here we will attempt to put such a definition on a sounder physical basis.

For most plants to grow in soil, a proportion of the pore space needs to be gas-filled. The gas-filled proportion of the pore space allows the influx of O2 into, and the efflux of CO2 from the soil. The consumption of O2 in the soil and production of CO2 is because of the oxidative demands of plant roots, microorganisms, and chemical reactions. The status of a soil in relation to the proportion of gas-filled pores and the concentration of O2 and CO2 in the soil is often referred to as the aeration status of the soil. The effect of aeration on plant growth has been reviewed recently by Glinski and Stepniewski (1985).

Although there appears to be some evidence to suggest that the concentration of CO2 in the soil may have an effect on plant growth, the major effect of poor aeration on plant growth is the lack of O2 (Glinski and Stepniewski, 1985). Like field capacity, soil aeration is a term that has an empirical basis and many different definitions. There are a number of different indices for defining the aeration status of a soil. These include: gas-filled porosity (Wesseling and van Wijk, 1957; Jayawardane and Meyer, 1985; Glinski and Stepniewski, 1985), the ODR (Lemon and Erickson, 1952; Letey and Stolzy, 1967; McIntyre, 1970; Armstrong and Wright, 1976; Blackwell, 1983), and O2 concentration or partial pressure (Grable, 1966; Armstrong and Gaynard, 1976; Blackwell and Wells, 1983; Meyer and Barrs, 1991). However, there has not been a good theoretical basis for investigating the validity of these indices for assessing the aeration status of a soil or their correlation. Traditionally, the concept of soil aeration has been based on a correlation between plant performance and these various indices of the O2 status of the soil.

At a physiological level, the minimum aeration status occurs when the flux of O2 to the root surface, especially the root tip, is just able to meet the O2 demand of the root tissues. The transport of O2 to a plant root occurs first through diffusion from the atmosphere via the gas-filled porosity of the soil to the depth of the root, and then through a boundary layer surrounding the root. Most previous studies in this area have considered this boundary layer around the root to be liquid-filled porosity, because as Bernstein et al. (1959) stated, otherwise the plant would have to have a refrigeration system if water was transported to it via a vapor gap. This liquid filled boundary layer was considered by Letey and Stolzy (1967) to be cylindrical. This assumption of a circular boundary layer was criticized by McIntyre (1970) as being highly unlikely given the heterogeneity of pore size found in soil (Fig. 1) . In this study the boundary layer will be assumed cylindrical, as without such an assumption the mathematics becomes very difficult. The value of this radius is taken as that which gives the same radial gas flux through the boundary layer as would be measured.



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  		Fig. 1. Schematic diagram of a root surrounded by an actual and hypothetical liquid-filled boundary layer, the radii a and R is also shown.

 
In this study we will derive the relationships between respiration rate of roots and the transport processes at the microscale (root scale) and couple this with the transport of O2 with root and microbial respiration as distributed sinks at the macroscale (bulk soil). The resulting solutions of the differential equations will be used to define soil aeration in terms of the physical and biological parameters of the system and to determine the interrelation of the three indices described above for defining soil aeration.


   THEORY
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
The transport of O2 to plant roots at the microscale can be described by radial diffusion to a cylindrical root (Letey and Stolzy, 1967):

[1]
where J is the flux of O2 to the root per unit length of root (kg m-1 s-1), {alpha} is the Bunsen absorption coefficient which equals the gas solubility per unit volume of water at normal pressure (101.3 kPa) (Glinski and Stepniewski, 1985 p. 46), C is the O2 concentration in the gas phase of the bulk soil (kg m-3), Cr is the O2 concentration in the liquid phase at the root surface (kg m-3), Dl is the diffusion coefficient of O2 in the liquid phase of the soil (m2 s-1), a is the root radius (m) and R is the radius of a zone of saturated soil around the root plus the root (m) (Fig. 1). Note Eq. [1] is different by 2{pi}a from that of Letey and Stolzy (1967), as here we have calculated the total O2 flux to the circular root surface, not the radial streamline flux. The relationship between ODR and Eq. [1] is algebraically possible but the physics is somewhat problematic as the boundary conditions are different. A good discussion of the ODR method and some of the problems with it are provided by McIntyre (1970) and Phene (1986).

The critical O2 concentration at the root surface can also be calculated using a radial solution for O2 transport into the root (Lemon and Wiegrand, 1962). Assuming that the concentration is zero at r = 0 (the center of the root), the concentration at the root surface is given by (Glinski and Stepniewski, 1985, p. 83):

[2]
where qr is the respiration rate of the root tissues (kg s-1 m-3) and Dr is the diffusion coefficient for O2 in the root (m2 s-1).

If we assume a root length density function, L (m m-3), with depth, z (m), of the exponential form:

[3]
where Lo is the root length density (m m-3) at z = 0 and Zr is a length scale (m), the root respiration per unit volume of soil (Qr) is then given by:

[4]

Assuming an exponential relationship with depth for the microbial respiration rate (Cook, 1995) the soil respiration per unit volume of soil (q) is given by adding the respiration rates:

[5]
where Mo is the microbial respiration rate at z = 0 (kg m-3 s-1) and Pz = is the root respiration rate where C(z) is the O2 concentration (kg m-3) in the soil air at depth z and Zm is a length scale (m). The length scales in both the root length density and microbial functions define the depth over which the respiration occurs. Ninety-five percent of the microbial respiration occurs within a depth of three length scales that is, z = 3Zm. We have not equated Zr and Zm as there are likely to be situations especially with a newly planted crop where they are not equal.

At the macroscale, one-dimensional O2 transport into a homogeneous soil can be described by diffusion (Kirkham, 1994). Steady-state diffusion into soil with the respiration considered as a distributed sink term is described by (Glinski and Stepniewski, 1985, p. 51, Eq. [39]):

[6]
where Da is the diffusion of O2 in the gas phase of the soil (m2 s-1). Substituting Eq. [5] into Eq. [6] results in:

[7]
which is to be solved with the following boundary conditions:

where A is the atmospheric concentration of O2 (kg m-3). The semi-infinite boundary condition used for the lower boundary condition is appropriate as the respiration tends toward zero with depth. Equation [7] is a second-order heterogeneous differential equation, which has the form:

[8]
where ß and g are parameters, with ß = , g = .

Introduction of a new spatial variable X = 2Zrg1/2exp, with the properties that when z = 0, X is a maximum X0 = 2Zrg1/2, and z = {infty}, X = 0. The derivative equals = -g1/2exp = -, which means that Eq. [8] can be rewritten as:

[9a]
or

[9b]
with the parameter p = , and the boundary conditions now:


A solution of the homogeneous equation with ß = 0 can be found in terms of the modified Bessel function Ip(X). When p is a positive integer a simple explicit particular solution of the heterogeneous equation can be found for each value of p. For general p, the solution can be written as an integral in Green's function form (see Appendix). When Zr = Zm and so p = 1, the solution with the appropriate boundary conditions is

[10]

The value at X = 0 is

[11]
since I0(0) = 1.

In terms of the original variables, in the case where Zr = Zm the solution is:

[12]

As z -> {infty}, this simplifies to:

[13]

It is via Da and Dl that the gas-filled porosity ({theta}) is introduced into the set of equations. Relationships between Dl and Da, and {theta} and soil porosity (f) are given by (Millington, 1959; Millington and Quirk, 1961):

[14]
where Doa is the diffusion coefficient for O2 in air and Dol is the diffusion coefficient for O2 in water. It can be shown that two distinct soil gas profiles are described by equations with exponentially decreasing sink terms; these are the case in which the O2 concentration becomes zero at some finite depth and the case when the O2 concentration tends toward a constant value as depth tends toward infinity (Cook, 1995). The soil profile will tend toward a constant O2 concentration in the air phase (C*) with depth as depth tends toward infinity when > and is given by Eq. [13]. Now if we let C* correspond to the critical value needed to satisfy the O2 flux to the plant roots then the rest of the soil profile will be at a value greater than this and soil aeration will be adequate throughout the soil profile. This is a similar concept to that introduced by Wesseling and van Wijk (1957) where they suggested that a {theta} of 10% at the bottom of the root zone was a critical index for soil aeration.

It can also be seen that via Eq. [1], [5], [13], and [14] the three common indices for determining aeration status viz constant values of either {theta}, O2 flux (this is related to ODR) and C are unified; if the parameters needed to calculate C* and {theta} are known, then O2 flux can also be calculated. The critical value for air-filled porosity ({theta}m) can be obtained from Eq. [13] using a binary splitting iterative procedure.

We now have a set of equations, which allows us to calculate the effects of variations in the soil physical, biological, and plant root characteristics on the aeration status of the soil. Examination of the above equations suggests that critical values for O2 concentration, oxygen flux, or air-filled porosity are unlikely to be universal, as in particular the temperature of the soil will vary in a growing season and diurnally and the sink terms will also vary in concert with temperature. The characteristics of the plant will also play a role in determining these critical values. The effects on {theta}m of varying the various soil and plant characteristics will be examined below. A critical O2 concentration (C*) will often be assumed, but this is not necessary. The critical O2 flux can be found with Eq. [1], using the assumed values of C* and Cr.


   MATERIALS AND METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
The critical air-filled porosity was calculated from Eq. [13] for typical values of the various parameters. A binary splitting procedure was written for Maple 7 (Waterloo Maple, 2001) and used to solve Eq. [13], so that both sides equated. Values for the various parameters were chosen from the literature, and are shown in Table 1. Each variable was then allowed to vary while the others were kept constant, to see how sensitive {theta}m was to variations in that parameter. These calculations were performed with the parameters calculated at a number of different temperatures. Variations in A, Doa, Dol, and {alpha} with temperature were taken from Glinski and Stepniewski (1985) and are shown in Table 1. Variation in Mo with temperature was calculated using the equation proposed by Lloyd and Taylor (1994):

[15]
where M is chosen so that the reference value of Mo is 3 x 10-7 kg m-3 s-1 at a temperature of 298 K, Eo is a constant (308.6 K) and To is a base temperature (227.1 K). The value of M was chosen as being representative of soil respiration values (Glinski and Stepniewski, 1985, Table 3, p. 35; Orchard and Cook, 1983).


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  		Table 1. Temperature dependence of variables (from Glinski and Stepniewski, 1985, p. 46).

 
The values of some of the temperature dependent variables at various temperatures are given in Table 1. These values were obtained from data in Glinski and Stepniewski (1985)(p. 46). The values of some parameters that remain constant with temperature during the sensitivity analysis are given in Table 2. The source of these values will be discussed below.


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  		Table 2. Constant values for nontemperature dependent variables used in sensitivity analysis.

 

   RESULTS AND DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
Concentration Profiles
The two different forms of an oxygen profile are seen in Fig. 2 ; when > , C tends toward a constant value of C with increasing depth. This is seen for the profiles when the temperature is at 273, 283, and 293 K in Fig. 2. When < , C goes to extinction at some finite depth, as is seen for the profile for a temperature of 303 K in Fig. 2. More detailed discussion of these profile shapes can be found in Cook (1995). In these calculations the temperature has been assumed to be uniform with depth. Temperature will vary with depth in soil under field conditions, although on a daily basis the amplitude is usually not large except close to the surface. However, the seasonal variation can be as much as 20 K, and these results show how this could affect the oxygen profiles for a soil on an annual basis.



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  		Fig. 2. Concentration of O2 in soil air (C) as related to depth in the soil, for different temperatures for a soil with {theta} = 0.13 m3 m-3. The other variables are as listed in Tables 1 and 2 and shown in Fig. 3.

 


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  		Fig. 3. Microbial (Mo) and root (Pz (z = 0) respiration terms as a function of temperature.

 
The assumption that Cr is constant with temperature is probably correct if the respiration processes in the root are concentration dependent. It is more likely that the root respiration and root diffusion coefficient vary with temperature. Some data on root respiration as a function of temperature has been published (Boone et al., 1998; Lemon and Wiegand 1962). The way in which this flux is likely to change with temperature is that the critical O2 concentration (C*) in the bulk soil will increase with temperature. However, no data on the variation of C* with temperature was found. Hence we have only allowed Pz to vary with temperature because of the variation of Dl and {alpha} with temperature. The temperature variations in Mo and Pz(z = 0) values are shown in Fig. 3 . Figure 3 also shows that the root respiration is greater than the microbial respiration, Glinski and Stepniewski (1985) suggested that root respiration is about ten-fold that of microbial respiration. The root respiration will however decrease with depth because of both the exponential forcing function and because of decreasing concentration of O2 with depth. How realistic Eq. [1] is without applying an upper limit to the flux, is something which requires further investigation outside the scope of this study. If, as is likely, root respiration varies with temperature, then the shape of the relationship between the Pz and temperature may be more like that shown for Mo in Fig. 3. This is likely to exacerbate the differences shown in Fig. 2 between the profile shapes at different temperatures, as the sink term (q) will vary more than assumed.

Sensitivity of {theta}m to Parameters in the Model
The sensitivity of {theta}m with respect to critical O2 concentration (C*) shows that as the critical concentration increases, so does the value of {theta}m (Fig. 4) . At the same value of C* the value of {theta}m increases with increasing temperature. The range of values chosen viz 0.025 to 0.20 kg m-3 is representative of values found by others (Glinski and Stepniewski, 1985, p14).



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  		Fig. 4. Sensitivity of {theta}m to C* for four different temperatures.

 
The definition of C* used here is the constant concentration in the soil when z > > Z. In this we are assuming that if C* is sufficient for root function at this depth then, since the concentration is greater than this in the upper part of the soil profile, it will be more than sufficient for root functioning. This definition is close in concept to a definition used by Wesseling and van Wijk (1957), except that their critical parameter was air-filled porosity. The values of {theta}m obtained range from 0.12 to 0.33 m3 m-3 for a soil with a porosity of 0.5 m3 m-3. This is similar to the range of values found in the literature of 0.1 to 0.25 (Glinski and Stepniewski, 1985. p. 173). Meyer and Barrs (1991) suggested a lower limit of 6% and an upper limit of 80% of pore space (f) to maintain a critical O2 concentration in the soil. For a porosity of 50% this would represent a {theta}m range of 0.03 to 0.4. The bottom limit of the range for {theta}m we found of 0.12 is higher than that found by others (Jayawardane and Meyer, 1985; Meyer and Barrs, 1991). This may have been because of the values of other variables including that of Cr. This value of Cr also restricts the lowest value of C* that can be used in the calculations as ({alpha}C* - Cr) > 0 is required in Eq. [2]. If Cr is allowed to vary with the value of C* constant (0.05 kg m-3) and the other variables constant at each temperature then the value of {theta}m hardly varies except when Cr > 1 x 10-4 Kg m-3 (Fig. 5) . The value of {theta}m then decreases slightly as Cr increases. This occurs because as Cr increases the flux of O2 to the root decreases and the total respiration decreases. These results suggests that {theta}m is not very sensitive to the value of Cr and hence to the values of qr, a, and Dr.



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  		Fig. 5. Sensitivity of {theta}m to Cr for four different temperatures.

 
The sensitivity of {theta}m to the strength of the microbial and root respiration is investigated via the parameters M, Lo, and Zm = Zr in Fig. 6 . Figure 6a shows that the root respiration rate dominates the sink terms until the microbial respiration term is about 1 x 10-6 kg s-1 m-3. At the lower values of M the root respiration is the dominant sink term, and results in convergence to a similar value of {theta}m at all temperatures. This occurs because of the lack of temperature sensitivity of the root respiration term discussed above.



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  		Fig. 6. Sensitivity of {theta}m to (a) Mo, (b) Lo and (c) Zm = Zr for four different temperatures.

 
Similarly, the microbial respiration dominates the total respiration at low values of Lo, and there is little change in {theta}m until high values of Lo are reached (Fig. 6b). At each temperature a relatively constant value of {theta}m occurs at values of Lo <= 1 x 103 m m-3. This is because of the microbial respiration being the dominant sink term at these low values of Lo. The temperature dependence of the microbial respiration (Fig. 3) is the reason why there is a different constant value for {theta}m at each temperature. At high values of Lo all the curves converge because of the lack of temperature dependence in the root respiration function.

When the value of Z is low, respiration is concentrated near the surface and the amount of oxygen consumed by the soil profile is lower. This results in a lower value of {theta}m (Fig 6c). These lower values (<0.1 m3 m-3) are similar to values found by Jayawardane and Meyer (1985). Such low values of {theta}m can also be obtained if both M and Pz are varied in concert, so that a wide range of values of the total respiration is created (Fig. 7) . Increasing the values of Z leads to an increase in {theta}m. This is to be expected, since the total O2 consumption of the soil profile will increase.



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  		Fig. 7. Sensitivity of {theta}m to qo (the total respiration at z = 0) at a temperature of 293 K.

 
The effect of root size and thickness of the water film surrounding the root shows that as R/a -> 1 the value of {theta}m increases. However for the range R/a > 10, {theta}m is relatively insensitive to R/a (Fig. 8) . The behavior of {theta}m as R/a -> 1 is because of the increase in flux of O2 to the root, which is assumed to happen in Eq. [1] as R/a -> 1. This results in an increase in the calculated root respiration and a corresponding increase in total respiration. The model then correctly suggests that {theta}m will increase as R/a -> 1. In reality this predicted increase in respiration is unlikely to occur, as respiration of the root will only rise to a maximum rate and then plateau, whereas Eq. [1] suggests it will increase exponentially as R/a -> 1. Also, the limit as R/a -> 1 corresponds to the water content approaching zero. The plant is likely to have stopped functioning and the plant roots stopped respiring well before R/a = 1 because of water stress.



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  		Fig. 8. Sensitivity of {theta}m to R/a for four different temperatures.

 
Equation [2] implies that Cr would be sensitive to the root radius a, but since {theta}m was shown to be insensitive to Cr (Fig. 5) it would appear from this analysis that {theta}m would not be sensitive to root radius. This analysis does not account for the fact that as water film thickness increases the water content increases and air-filled porosity decreases. Luxmoore et al. (1970) did attempt to relate water film thickness to soil matric potential using an analysis of water films on clays (Kemper and Rollins, 1966) to predict the water film thickness. Unfortunately their work is erroneous, as they made an order of magnitude error in converting from the units of A° (10-10 m) used by Kemper and Rollins (1966) to their units of centimeters.

Increasing the total porosity f is shown to increase {theta}m almost linearly (Fig. 9) . However, the proportion of the porosity represented by {theta}m decreases as f increases (not shown). This almost linear decrease is because of the term ß/g decreasing with f but being compensated for by the decrease in the modified Bessel function with f. This result is dependent on the diffusion function of Millington and Quirk (1961) being a good representation of reality. Sallam et al.(1984) showed that Millington and Quirk's function describes the diffusion of gases in soil well, but suggested that the power on the {theta} term be changed from 10/3 to 3.1. In the calculations presented here, we have not bothered with this minor variation.



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  		Fig. 9. Sensitivity of {theta}m to f for four different temperatures.

 
The sensitivity of {theta}m to temperature is shown in Fig. 4 to 9. As the temperature increases so does the value of {theta}m required to provide a value of C* >= 0.05 kg m-3 in the soil profile. This indicates that the temperature is a vital parameter that should be measured in any experimental work on aeration.

Comparison with Other Values of Critical Parameters
Critical values of the ODR (Letey and Stolzy, 1967; Blackwell and Wells, 1983) or O2 flux rate (Blackwell and Wells, 1983) have been suggested, and these vary from 150 to 9 µg m-2 s-1 (see Glinski and Stepniewski, 1985, Fig. 20, p. 80). From the analysis presented here it is possible to investigate whether a universal value of O2 flux is likely to exist. The parameters in Eq. [1] are {alpha}, Dl, C*, Cr, a, and R. The value of {alpha} decreases by about 50% between temperatures of 273 and 303 K, as shown in Table 3 of Glinski and Stepniewski (1985)(p. 46). Values of Dl will vary with both f and temperature. The parameter Cr may vary with temperature (Lemon and Wiegand, 1962), while C* is likely to vary with temperature and plant species. There is also likely to be an effect because of mycorrhizal microorganisms, as their respiration rate will affect the value of C*. The parameter a is likely to vary with plant species, while R is likely to vary with soil type. Hence it is not surprising that suggested critical values vary over a wide range. It is doubtful if a universal value for a critical value of ODR exists. If such a value does exist it can be shown using Eq. [1], [13], and [14] that constant critical values for C* and {theta}m cannot exist. Similarly, if a constant value for C* exists then a constant value of O2 flux or {theta}m cannot exist. The respiration rate of root tissues has been found to be exponentially related to temperature (Boone et al., 1998) like microbial respiration. Yet for the existence of a constant critical flux to be correct the respiration would have to be constant with temperature. It is difficult to assess the data on O2 flux or ODR, as the temperature at which measurements were made is not given often.

Critical values for the O2 concentration in the gas or water phase have been proposed (Blackwell and Wells, 1983; Glinski and Stepniewski, 1985; Meyer and Barrs, 1991), and these values also vary over a wide range. There is evidence that a constant root O2 concentration is required for proper functioning of cytochrome oxidase (Glinski and Stepniewski, 1985, p. 141), and Glinski and Stepniewski (1985)(p. 83) suggest that this constant root O2 concentration is not <0.02 to 0.03 m3 m-3. Equation [1] then shows that, if Cr is constant and J is constant, then, as Dl and {alpha} are temperature dependent and Dl is dependent on f, C* cannot be constant. Thus, the existence of a critical constant flux and the existence of a constant critical concentration are mutually exclusive. At a single temperature, as in the measurements of Blackwell and Wells (1983) at 281 K, a correlation can be found, but at different temperatures this relationship is likely to be different.

Meyer and Barrs (1991) suggested that the O2 in both the air and water phases needs to be taken into account in determining a critical value (Cc) of 0.1 kg O2 m-3:

[16]
where {theta} (m3 m-3) is the air-filled porosity. This concept does not take into account that the water phase is merely a conductor for O2 from the gas phase to the root. Thus under steady-state flow conditions the concentration of O2 in the water phase must stay constant and the flux from the atmosphere to the soil will occur predominantly in the gas phase. However, if roots have a diurnal variation in their O2 uptake rate and the amount stored in the water phase is sufficient, then there may be some justification for the suggestion of Meyer and Barrs (1991). We have calculated the time it would take to deplete the O2 in the water phase, at the maximum respiration rate (z = 0) and given no replenishment from the gas phase, for a range of values of (f - {theta}a) and at various temperatures. This shows that at low temperatures and high water contents (f - {theta}a) there is a considerable length of time that O2 could be supplied from that dissolved in the liquid phase (Fig. 10) . Also, at lower respiration rates than assumed here this time will increase. This would indicate that the suggestion of Meyer and Barrs (1991) might be valid when the O2 demand of the soil is low, if diurnal fluctuations in the respiration rate of microbes and roots occur, allowing replenishment of O2 in the liquid phase during the period of low respiration. Although there may be a universal critical value of Cr (Glinski and Stepniewski, 1985 p. 141), a universal value of C* is highly unlikely. The sensitivity analysis here has used a constant value of 0.05 kg m-3 but the trends in the analysis are similar if other values are chosen.



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  		Fig. 10. The time to deplete the oxygen dissolved in the water phase, as related to the water content of the soil (f- {theta}a) at four different temperatures. This analysis assumes no replenishment of the oxygen in the water phase during this depletion and that the root respiration was calculated with C = 0.05 kg m-3.

 

   DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 DISCUSSION
CONCLUSIONS
 APPENDIX
 REFERENCES
 
For soil aeration studies to move beyond empirical statistical correlations between plant response and O2 flux or C* or {theta}m, it is necessary to make experimental measurements that allow the model presented here to be tested. The soil respiration, length scale, and O2 concentration can be derived using profile methods (Cook et al., 1998) and the analysis of Cook (1995). Concurrent measurements of temperature profiles and water content are also required (Topp et al., 2000). The splitting of the total respiration into the microbial and root components is more problematic. Studies of the roots in isolation may be possible using the agar gel method of Wiengweera et al., (1997). It is possible to sieve out the roots and measure the microbial respiration rate (Orchard and Cook, 1983), but these values may not be a good measure of the in situ value.

In this study we have not included the effect of water content on microbial respiration. It has been shown that water content has a marked effect on microbial respiration (Orchard and Cook, 1983; Linn and Doran, 1984). Orchard and Cook (1983) only studied the range of water contents where O2 was not limiting and found a linear decrease with water content. This relationship was confirmed in later studies (Cook et al., 1985; Orchard et al., 1992), whereas Linn and Doran (1984) showed that above a critical value the respiration rate decreases with water content because of lack of O2. As we are looking for point of maximum respiration here, neglecting the effect of water content on respiration will not affect the results. However, if Eq. [12] is used to predict O2 profiles the effect of water content on microbial respiration will need to be accounted for.


   CONCLUSIONS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 DISCUSSION
 CONCLUSIONS
APPENDIX
 REFERENCES
 
A theoretical model for aeration has been presented, showing that universal critical values for the air-filled porosity, O2 concentration and ODR are unlikely to exist. The model results also imply that if a universal critical value exists for one of these parameters it cannot exist for the other critical parameters.

The results show that when a critical value for the air-filled porosity is used as an indicator the model is not very sensitive to the concentration on the root surface (Cr), but is sensitive to all other parameters. The model is sensitive to some of the variables, R/a, Lo, and M only in certain ranges.

This study shows that for progress to be made in understanding soil aeration a much larger suite of parameters needs to be measured. Some of these can be obtained by measurements of O2 flux, soil O2, water content, and temperature.


   APPENDIX
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 DISCUSSION
 CONCLUSIONS
 APPENDIX
REFERENCES
 
The equation for C(X) for a general value of p is

[A1]

For p = 1, the case when the two characteristic lengths Zm and Zr are equal, this has the form

[A2]

This has a particular solution C = that is used in Eq. [10].

The case p = 2 corresponds to the relation between the two characteristic lengths Zr = 2Zm and the equation has the form

[A3]
with particular solution

[A4]

For p = 2 the solution satisfying the boundary conditions, C = A - , X -> 0 as X -> 0 is

[A5]

For each integer value of p a particular solution can be found as a finite series of powers of X.

For general p a solution can be written down in terms of the integral of a Green's function as

[A6]
where the Green's function is a solution of the homogeneous equation with a "delta function" unit source term at the position X = Y. For this equation the Green's function satisfying the appropriate boundary conditions is

[A7]

where Ko is a modified Bessel function of second kind and zero order. For integer values of p the integrals can be evaluated explicitly, after some manipulation of the Bessel functions.

List of Symbols
a root radius (m)

A O2 concentration in the atmosphere (kg m-3)

C O2 concentration in gas phase of soil (kg m-3)

C* critical O2 concentration in gas phase of soil as z -> {infty} (kg m-3)

Cr critical O2 concentration in liquid phase at root surface (kg m-3)

Da diffusion coefficient for O2 in soil through the gas phase (m2 s-1)

Doa diffusion coefficient for O2 in air (m2 s-1)

Dl diffusion coefficient for O2 in soil through the water phase (m2 s-1)

Dol diffusion coefficient for O2 in water (m2 s-1)

Dr diffusion coefficient for O2 in root tissues (m2 s-1)

Eo constant analogous to activation energy (K)

f porosity of soil (m3 m-3)

g parameter defined in Eq. [8] (kg m-5)

G Green's function (see Eq. [A6])

Io modified Bessel function of first kind and zero order

J flux of O2 to the root per unit length of root (kg m-1 s-1)

Ko Bessel function of second kind and zero order

L root length density (m m-3)

Lo root length density at soil surface (m m-3)

M reference microbial respiration rate (kg m-3 s-1)

Mo microbial respiration rate at soil surface (kg m-3 s-1)

Pz root respiration rate at depth z (kg m-3 s-1)

p dimensionless parameter defined in Eq. [9]

q total soil respiration per unit volume of soil (kg m-3 s-1)

qr respiration (consumption of O2) rate per unit length of root (kg m-1 s-1)

Qr respiration (consumption of O2) rate per unit volume of root (kg m-3 s-1)

R radius of root plus water film thickness (m)

T temperature (K)

To base temperature (K)

X spatial variable used in Eq. [9] (kg1/2 m-1/2)

Xo boundary condition for X (kg1/2 m-1/2)

Y is a dummy variable used in Eq. [A6]

z depth (m)

Zm length scale for microbial respiration function (Eq. [5]) (m)

Zr length scale for root length density function (Eq. [3]) (m)

Z length scale used when Zm = Zr (m)

{alpha} Bunsen coefficient (m3 m-3)

ß parameter defined in Eq. [8] (kg m-5)

{theta} air-filled porosity (m3 m-3)

{theta}m critical air-filled porosity (m3 m-3)


   ACKNOWLEDGMENTS
 
We acknowledge improvements to this manuscript suggested by Drs. M.B. Kirkham and J.B. Passioura, and the encouragement of Drs. I. White, G.C. Topp, and J. Williams to commence and complete this work. Lesley Cook is thanked for her encouragement, patience, and support during the many evenings spent on the calculations.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 
Contribution from CSIRO Land and Water.

Received for publication December 11, 2000.


   REFERENCES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 DISCUSSION
 CONCLUSIONS
 APPENDIX
 REFERENCES
 




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