An analytical solution for steady-state oxygen transport in soils including 2 sink terms, viz roots and microbes with the corresponding vertical distribution scaling lengths forming a ratio p, showed p governed the critical air-filled porosity, θ_c, needed by most plants. For low temperature and p, θ_c was <0.1 but at higher temperatures and p = 1, θ_c was >0.15 m³/m³. When root length density at the surface was 10⁴ m/m³ and p > 3, θ_c was 0.25 m³/m³, more than half the pore space. Few combinations of soil and climate regularly meet this condition. However, for sandy soils and seasonally warm, arid regions, the theory is consistent with observation, in that plants may have some deep roots. Critical θ_c values are used to formulate theoretical solutions in a forward mode, so different levels of oxygen uptake by roots may be compared to microbial activity. The proportion of respiration by plant roots increases rapidly with p up to p ≈2.

Synthesis of vertical root biomass density, L [= L₀ exp(–z/Z_r), z is the depth positive down (m)] (m/m³), data using an exponential function to represent the distribution suggested that, on average, 70 ± 10% of fine roots in 10 terrestrial biomes were located in the upper 0.1 m of soil. Integrated over the root-zone, L_T is given by the product of the function’s 2 parameters, the surface value of L, L₀ (m/m³), and length scale, Z_r (m). As postulated, negative correlations were obtained between L₀ and Z_r. For a maize (Zea mays L.) crop, significantly different distributions were measured during relatively dry and wet seasons and predicted by our model. For woody and herbaceous plants, Z_r (the value determines the rate of decrease in L with depth) averaged 0.3 and 0.2 m, respectively, while the corresponding averages for R_m0 [= L₀.ρ_r, ρ_r is root density (kg/m)] were 2.7 and 1.1 kg/m³.