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A boundary-layer solution for flow at the soil-root interface

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Abstract

Transpiration, a process by which plants extract water from soil and transmit it to the atmosphere, is a vital (yet least quantified) component of the hydrological cycle. We propose a root-scale model of water uptake, which is based on first principles, i.e. employs the generally accepted Richards equation to describe water flow in partially saturated porous media (both in a root and the ambient soil) and makes no assumptions about the kinematic structure of flow in a root-soil continuum. Using the Gardner (exponential) constitutive relation to represent the relative hydraulic conductivities in the Richards equations and treating the root as a cylinder, we use a matched asymptotic expansion technique to derive approximate solutions for transpiration rate and the size of a plant capture zone. These solutions are valid for roots whose size is larger than the macroscopic capillary length of a host soil. For given hydraulic properties, the perturbation parameter used in our analysis relates a root’s size to the macroscopic capillary length of the ambient soil. This parameter determines the width of a boundary layer surrounding the soil-root interface, within which flow is strictly horizontal (perpendicular to the root). Our analysis provides a theoretical justification for the standard root-scale cylindrical flow model of plant transpiration that imposes a number of kinematic constraints on water flow in a root-soil continuum.

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Acknowledgments

This research was supported in part by the National Science Foundation award EAR-1246315 and by the Computational Mathematics Program of the Air Force Office of Scientific Research. The first author acknowledges support from “Programma di scambi internazionali per mobilitá di breve durata” (Naples University, Italy), “OECD Cooperative Research Programme: Biological Resource Management for Sustainable Agricultural Systems” (Contract No. JA00073336), and PRIN project “I paesaggi tradizionali dell’agricoltura italiana: definizione di un modello interpretativo multidisciplinare e multiscala finalizzato alla pianificazione e alla gestione” (Contract No. 2010LE4NBM_007). The first author thanks Prof. Gerardo Toraldo (Naples University) for promoting his visit to University of California, San Diego; and Dr. Peng Wang for his kind hospitality.

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Authors and Affiliations

  1. Division of Water Resources Management and Bio-System Engineering, University of Naples, Federico II via Universitá 100, 80055 , Portici, Naples, Italy

    Gerardo Severino

  2. Department of Mechanical and Aerospace Engineering, University of California, 9500 Gilman Drive EBU II, Mail Code 0411 La Jolla, San Diego, CA, 92093, USA

    Daniel M. Tartakovsky

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  1. Gerardo Severino
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  2. Daniel M. Tartakovsky
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Correspondence to Gerardo Severino.

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Severino, G., Tartakovsky, D.M. A boundary-layer solution for flow at the soil-root interface. J. Math. Biol. 70, 1645–1668 (2015). https://doi.org/10.1007/s00285-014-0813-8

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  • DOI: https://doi.org/10.1007/s00285-014-0813-8

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