*b* | Buffer power of soil^{1}, (A3) | 5 |

*C* _{s}(*r*), *C*_{s}*′* | Solute concentration at radial distance r from the root surface, dimensionless solute concentration, (A3), (A4) | mol N cm^{−3}, - |

*C* _{o}, *C*_{o}′ | Solute concentration at the root surface, dimensionless solute concentration at the root surface, (A1), (A8) | mol N cm^{−3}, - |

*CRP*(*z*) | Empirical function for cumulative root proportion to depth *z*^{2,3}, (15) | - |

*D* _{max} | Maximum rooting depth, (4) | m |

*D* _{o} | Length scale for exponential decline of *U*_{o} with depth^{4}, (2) | 0.3 m |

*E* | Daily water extraction by roots from unit soil volume, (A3) | cm^{3} water cm^{−3} soil volume day^{−1} |

*F*(*z*) | Empirical function for root-depth distribution^{5}, (14) | m^{−1} |

*I* | Rate of N uptake by root per unit root surface area^{6}, (A1) | mol N cm^{−2} root surface day^{−1} |

*L* _{r}(*z*), *L*_{r}*′* | Root-length density at depth *z*, dimensionless root-length density, (A1), (A6) | cm root cm^{−3} soil volume, – |

*L* _{ro} | Root-length density at half maximum potential N uptake, (1) | 0.77 cm^{−2} |

| Dimensionless root-length density at half-maximum potential N uptake, (A9) | - |

*N* _{r} | Nitrogen concentration of fine roots^{7}, (3) | 6.8 g N (kg DM)^{−1} |

*r, r′* | Radial distance from centre of root, dimensionless radial distance, (A3), (A5) | cm, - |

*r* _{o}, *r*_{o}′ | Fine-root radius^{8}, dimensionless root radius, (A1), (A7) | 0.017 cm, - |

*r* _{x}, *r*_{x}′ | Inter-root distance^{6}, dimensionless inter-root distance, (A2), (A7) | cm, - |

*R*(*z*) | Root mass per unit soil volume, or root-mass density at depth *z*, (1) | kg DM m^{−3} |

*R* _{av} | Average root-mass density over the rooting zone, (A21) | kg DM m^{−3} |

*R* _{o} | Root-mass density at half maximum potential N uptake, (1) | 0.265 kg DM m^{−3} |

*R* _{tot} | Total root biomass per unit land area^{7,8}, (6) | kg DM m^{−2} |

*u* _{maxi} | Potential N uptake per unit land area on day *i*, (A31) | g N m^{−2} day^{−1} |

*u* _{oi}(*z*) | Potential N uptake per unit soil volume at depth *z* on day *i*, (A29) | g N m^{−3} day^{−1} |

*u* _{ri}(*z*) | N uptake per unit soil volume at depth *z* on day *i*, (A29) | g N m^{−3} day^{−1} |

*U* | N uptake per unit soil volume derived from the Barber–Cushman model, (A1) | mol N cm^{−3} day^{−1} |

*U* _{o} | Potential annual N-uptake rate per unit soil volume^{9}, (1), (A3) | g N m^{−3} year^{−1}(Main text, Appendices A2 and A3), mol N cm^{−3} day^{−1} (Appendix A1), |

*U* _{o}(*z*) | Potential annual N uptake per unit soil volume at depth *z*, (1) | g N m^{−3} year^{−1} |

*U* _{n}(*z*) | Annual N supply to aboveground pools per unit soil volume, (3) | g N m^{−3} year^{−1} |

*U* _{r}(*z*) | Annual total N uptake per unit soil volume, (1) | g N m^{−3} year^{−1} |

*U* _{max} | Potential annual N uptake per unit land area integrated over the soil profile ()^{9}, (2) | 13.6 g N m^{−2} year^{−1} |

*U* _{net} | Annual N supply to aboveground pools per unit land area^{7,10}, (4) | g N m^{−2} year^{−1} |

*U* _{tot} | Annual total N uptake per unit land area^{7,11}, (5) | g N m^{−2} year^{−1} |

*z* | Soil depth, (1) | m |

*Z* _{o} | Length scale for the exponential decline of empirical root distribution with depth^{5}, (14) | m |

α, α′ | Root absorbing capacity^{1,6}, dimensionless root absorbing capacity , (A1), (A8) | 5.33 cm day^{−1}, - |

β | Exponent in the empirical relationship for cumulative root proportion^{2,3}, (15) | 0.914 (tundra), 0.972 (aCO_{2}), 0.984 (eCO_{2}) |

Δ | Diffusion coefficient of nutrient in soil^{1}, (A3) | 0.052 cm^{2} day^{−1} |

φ_{N}, φ_{net} | Gross N-uptake fraction (=*U*_{tot}/*U*_{max}) and net N-uptake fraction (=*U*_{net}/*U*_{max}), (12), (13) | - |

φ, φ | Peak values of φ_{N} and φ_{net}, (A28), (A27) | - |

λ | Lagrange multiplier ∂*U*_{n}/∂R, (A10) | g N kg^{−1} DM year^{−1} |

µ | Rate of solute loss from the rhizosphere through immobilisation by soil microbes^{12}, (A3) | day^{−1} |

ρ_{r} | Root-tissue density^{8}, (1) | 380 kg DM m^{−3} |

ξ | Expression , (13) | - |

τ_{r} | Root lifespan^{8}, (3) | 1 year |

Ψ | Goal function, (A10) | g N m^{−2} year^{−1} |

ζ | Expression , (A25) | - |