J. Math. Biol. (2015) 70:1645–1668DOI 10.1007/s00285-014-0813-8 Mathematical Biology

A boundary-layer solution for flow at the soil-rootinterface

Gerardo Severino · Daniel M. Tartakovsky

Received: 17 July 2013 / Revised: 16 June 2014 / Published online: 10 July 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract Transpiration, a process by which plants extract water from soil and trans-mit it to the atmosphere, is a vital (yet least quantified) component of the hydrologicalcycle. We propose a root-scale model of water uptake, which is based on first princi-ples, i.e. employs the generally accepted Richards equation to describe water flow inpartially saturated porous media (both in a root and the ambient soil) and makes noassumptions about the kinematic structure of flow in a root-soil continuum. Using theGardner (exponential) constitutive relation to represent the relative hydraulic conduc-tivities in the Richards equations and treating the root as a cylinder, we use a matchedasymptotic expansion technique to derive approximate solutions for transpiration rateand the size of a plant capture zone. These solutions are valid for roots whose sizeis larger than the macroscopic capillary length of a host soil. For given hydraulicproperties, the perturbation parameter used in our analysis relates a root’s size to themacroscopic capillary length of the ambient soil. This parameter determines the widthof a boundary layer surrounding the soil-root interface, within which flow is strictlyhorizontal (perpendicular to the root). Our analysis provides a theoretical justificationfor the standard root-scale cylindrical flow model of plant transpiration that imposesa number of kinematic constraints on water flow in a root-soil continuum.

G. Severino (B)Division ofWater ResourcesManagement andBio-SystemEngineering,University ofNaples, FedericoII via Universitá 100, 80055 Portici, Naples, Italye-mail: severino@unina.it

D. M. TartakovskyDepartment of Mechanical and Aerospace Engineering, University of California, 9500 Gilman DriveEBU II, Mail Code 0411 La Jolla, San Diego, CA 92093, USAe-mail: dmt@ucsd.edu

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1646 G. Severino, D. M. Tartakovsky

Keywords Plant transpiration ·Water uptake · Rhizosphere · Singular perturbation ·Matching asymptotic

Mathematics Subject Classification 76S05 · 92C80 · 34D15

1 Introduction

Transpiration is widely recognized to be a fundamental component of the hydrologicalcycle. It is also one of the least quantified, typically relying on empirical constitutivelaws (e.g. Green et al. 2006). Fluid mechanics of transpiration involves coupled non-linear flows in two adjacent media, a root system and an ambient soil (see the compre-hensive reviews by Rand 1983; Passioura 1988; Green et al. 2006). These flows canbe described by the Richards equation (e.g. Warrick 2003; Sperry et al. 2002),

∂θ

∂t= −∇ · q, q = −K(ψ)∇�, (1)

where θ ≡ θ(x, t) is the volumetric water content of a medium, K ≡ K(ψ) is thehydraulic conductivity tensor,q(x, t) is themacroscopicDarcy flux,�(x, t) is the totalhydraulic head, and a constitutive law (retention curve) relates the two state variables θ

and�. The hydraulic head of water in a soil,� = ψ + z, consists of the pressure head(also known as matric potential or suction) ψ = pw/γw (with pw and γw denotingthe pressure and the specific weight of water, respectively) and the elevation head z.Pressure head in partially saturated media is less than atmospheric pressure head, i.e.ψ < 0.

In roots, the gradient of osmotic tension π can act as an additional force (Fiscus1975), giving rise to � = ψ + z − σπ where σ is a reflection coefficient for solutes.However, it has long been argued (e.g. Fiscus 1975; Passioura 1988) that its influenceduring periods of active transpiration is negligible because soil water is usually diluted(Weatherley 1982). Following Roose and Fowler (2004a), Roose and Fowler (2004b),Schneider et al. (2010) and many others, we assume that σπ � ψ + z, i.e. define thetotal hydraulic head of water throughout a soil-root system as � = ψ + z. This isappropriate for transpiration of low-salinity soil water (Green et al. 2006).

While a plethora of well-established constitutive laws and measurement techniquescan be used to describe the pressure-dependent hydraulic conductivity of soils K (ψ)

(e.g. Warrick 2003), practical limitations of dealing with live plants render in situdetermination of K (ψ) for roots more problematic (Steudle 2000). Generally, the rootconductivity depends on plant species, their age, and temperature (Lopez and Nobel1991; Tsuda and Tyree 2000, and the references therein ). Germane to this study isan experimental observation (ibid) that root conductivities decrease in drying soils,causing reduction in the water flux through a root.

Modern quantitative understanding of water uptake by roots dates back to 1950–60s mesoscopic models, which (1) treat a root as an infinite cylinder (of radius r0)embedded in an infinite soil cylinder (of radius r1, such that r0 � r1); (2) assume thatflow from the soil into the root is strictly horizontal, i.e. employ the one-dimensional(in the radial direction) version of (1) to model flow in the soil shell r0 < r < r1;(3) suppose that the horizontal flow is driven by the difference in matric potentials

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Flow at the soil-root interface 1647

ψ0 and ψ1 at the boundaries r = r0 and r = r1, respectively; and (4) impose eithera given flux (for constant rate of uptake) or a given saturation (for falling rate ofuptake) at the root-soil interface r = r0 (Raats 2007, and the references therein ). Akey finding of these models is that water uptake at both meso- and macro-scales islinearly proportional to the head difference �1 − �0 (Throughout this study we useRaat’s definition of scales: “At the mesoscopic scale, uptake of water is representedby a flux at the soil-root interface, while at the macroscopic scale it is represented by asink term in the volumetric mass balance.”). This result underpins more sophisticatednumerical models of plant transpiration and root water uptake (Roose and Fowler2004a, b; Jong-van-Lier et al. 2006; Metselaar and Jong-van-Lier 2007; Javaux et al.2008; Schneider et al. 2010; Couvreur et al. 2012).

We propose a root-scale analytical model, which either relaxes or eliminates theassumptions mentioned above. Instead of using an infinite cylinder to conceptualize aroot, it imposes geometric and hydraulic constraints on its length. Our model does notpostulate the existence of a soil cylinder with a known and spatially constant hydraulichead prescribed on its surface, and accounts for the vertical component of flow velocityin both the root and the soil. It uses measurable hydraulic quantities (infiltration rateand root-system suction) as input parameters, and enforces fundamental conservationlaws at the root-soil interface. In Sect. 2, we formulate a mathematical model of waterflow in soil-root systems and discuss its assumptions and limitations. In Sect. 3, weemploymatched asymptotic expansions to derive an analytical solution to this problem.Its physical implications are discussed in Sect. 4. Salient features of our analysis aresummarized in Sect. 5.

2 Problem formulation

2.1 Flow domain

A root system of a typical plant is geometrically complex, forming non-uniformbranching networks. The analysis presented below explores kinematic underpinningsof the commonly used “cylindrical flow model” formulated as follows (Passioura1988),

“Despite the fact that root systems are branched, that the catchments of individ-ual roots overlap in geometrically complicated ways, and that roots growing inreal soil are typically not cylindrical because they must weave their ways pastobstructions and often have to conform to the shapes of the pores within whichthey are growing, this model [water flowing radially towards a cylindrical, essen-tially isolated, root] remains useful. It can be extended to complete root systems,despite their complexity, by means of the simple but effective stratagem of imag-ining that each root has exclusive access to a cylinder of soil whose outer radius,Reff , is half the average distance between roots. Radius Reff can be calculatedas Reff = 1/

√πρav, where ρav is the average rooting density, the length of root

in unit volume of soil.”

From the outset, it is important to recognize that what Passioura (1988) and much ofthe subsequent literature on the subject call “the cylindrical flow model” refers not

123

1648 G. Severino, D. M. Tartakovsky

q

water table

soil surface

qq

r

z

L

2RB

L

2R

Fig. 1 Sketch of the water uptake induced by plant transpiration: a root system and its conceptualizationwith a single cylindrical root of radius R and length L . The vadose zone of thickness B separates the soilsurface from the water table. Constant infiltration rate q is prescribed at the soil surface

only to root geometry but also to the flow configuration, in which water flows “radiallyin response to gradients in pressure, towards a cylindrical, essentially isolated, root”.

We generalize the classical cylinder flow model by obviating the need for a priorirestrictions on flow configuration in either the root system or the ambient soil. Weconsider a single root of radius R and length L in a homogeneous vadose zone (Fig. 1).The depth to groundwater, i.e. the thickness of a partially saturated soil (vadose zone),is denoted by B and assumed to be large enough for flow around the root to beunaffected by the water table. Finally, we assume axial symmetry and place the outerboundary of the flow domain at r = Reff , such that Reff � R as defined in Passioura(1988). A baseline average rooting density ρav = 1.0 cm cm−3 (Passioura 1988)would result in the outer radius Reff ≡ 1/

√πρav ∼ O(1 cm). The resulting flow

domain = {(r, z) : 0 ≤ r < Reff , 0 ≤ z < B} consists of two subdomains, = x ∪ s where x = {(r, z) : 0 ≤ r < R, B − L ≤ z < B} and s = /x

represent the root (xylem) and ambient soil, respectively. (Table 1 lists these and othersymbols used in our analysis.)

2.2 Flow in ambient soil

Meteorological conditions that predominantly control transpiration rates vary rapidlyin time, exhibitingmarked diurnal fluctuations. Yet, a large number of studies reviewed

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Flow at the soil-root interface 1649

Table 1 Summary of the physical quantities and their dimensional counterparts

Symbol Quantity Units

(r, z) Radial and vertical coordinates L

B Vadose zone thickness L

L Root length L

R Root radius L

Reff External radius of the flow domain L

ψ Pressure head L

ψ0 Prescribed root suction at z = B L

q = (qr , qz) Macroscopic (Darcy) flux L/T

q Infiltration rate at soil surface L/T

K0 & Kx Saturated hydraulic conductivities of xylem L/T

in the r and z Directions, respectively

Ks Saturated hydraulic conductivity of ambient soil L/T

αx Pore-size distribution parameter of xylem 1/L

αs Pore-size distribution parameter of ambient soil 1/L

� Modified Kirchhoff transform of ψ L

α Step function equal to αx in root and αs in soil 1/L

κr , κr , κz & κz Conductivity ratios defined in (12b) –

K = K0/Kx Conductivity anisotropy ratio in xylem –

χ = αs/αx Distribution parameter ratio –

�r , �z , �� Characteristic lengths defined in Sect. 3.2 –

ε = (�z/�r )2 Small perturbation parameter –

A Dimensionless counterparts of quantities A = r, z, . . . –

(r , z) Rescaled coordinates in boundary layer –

R = αx R/2 Scaled root radius –

J � Computed average transpiration flux –

by Raats (2007) suggest that flow in both root systems and ambient soils can berepresented by a sequence of steady states. This implies that a perturbation in thewater potential ψ propagates instantaneously through a root zone. Cowan (1965),and the subsequent theoretical and experimental studies reviewed by Raats (2007),demonstrated that this approximation is adequate under typical conditions. Steady-state solutions are also used to interpret experimental data for water uptake by rootsof various plants (Raats 2007, Sect. 4.5).

We therefore consider a steady-state (∂θ/∂t = 0) version of (1) with � = ψ + z,

∇ · [Krel(ψ)∇ψ] + ∂

∂zKrel(ψ) = 0 (r, z) ∈ s . (2)

This formulation implies that the soil is homogeneous and isotropic, and representsits hydraulic conductivity K (ψ) = KsKrel(ψ) as the product of (constant) saturatedhydraulic conductivity Ks and the pressure-dependent relative hydraulic conductivity

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1650 G. Severino, D. M. Tartakovsky

Krel(ψ). For the latter, we choose Gardner’s model (e.g. Warrick 2003),

Krel(ψ) = eαsψ, (3)

where αs is the pore-size distribution parameter. Due to its relative simplicity, theexponential model (3) is often used in both theoretical and experimental investigationsof flow in partially saturated porous media. Tartakovsky et al. (2003a, 2004) providean extensive list of studies that employ Gardner’s model. Our choice of Gardner’smodel (3) facilitates the subsequent analytical treatment.

While more complicated constitutive relations, a plethora of which are compiled inSection 2.5 of Warrick (2003), often fit conductivity vs. saturation (or pressure head)data better, the reliance on the Gardner model is justified for the following reasons.First, using such constitutive relations (e.g. the Brooks-Corey or van Genuchten mod-els) to fit sparse, spatially heterogeneous and error-prone data might not be warrantedfrom the information-theory point of view. That is because these relations invariablerely on two or more fitting parameters, whereas Gardner’s model (3) has only one.Second, the differences between theGardnermodel and, say, the vanGenuchtenmodelare usually limited to saturation extremes corresponding to either nearly dry or nearlycompletely saturated soils, both of which detrimental to many plants. Third, the oper-ational equivalency between various constitutive models Krel(ψ) is achieved throughtheir parameterization in a way that preserves themaximum value of macroscopic cap-illary length, which is defined as Hc = ∫ ∞

0 Krel(s)ds (e.g. Tartakovsky et al. 2003a).This quantity, which is also referred to as effective capillary drive, is directly relatedto the Kirchhoff transform used in the analysis below.

External boundary conditions for the Richards equation (3) are determined byphysical processes at the soil surface (z = B) and water table (z = 0). Let q denotea prescribed (negative for infiltration) vertical Darcy flux at the soil surface z = B(Fig. 1) due to rainfall or irrigation. This gives rise to a boundary condition

eαsψ

(

1 + ∂ψ

∂z

)

= − q

Ks, z = B, R < r < Reff . (4a)

The water table (z = 0) separates the (partially saturated) vadose zone wherepressure head ψ < 0 from the (fully saturated) phreatic zone where ψ > 0. Hence, aboundary condition at the water table reads

ψ(r, 0) = 0, 0 � r < Reff . (4b)

This formulation ignores (without loss of generality of the present analysis) the cap-illary fringe and sets atmospheric pressure head to ψatm = 0. Finally, the externalboundary r = Reff is located sufficiently far from the root as to ensure that the latterdoes not affect the vertical flow (infiltration) in the rest of the soil (r ≥ Reff ). Thecorresponding boundary condition is

∂ψ

∂r(r = Reff , z) = 0, 0 ≤ z ≤ B. (5)

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Flow at the soil-root interface 1651

2.3 Flow in root xylem

While soils might or might not be anisotropic, xylem’s morphology renders itshydraulic conductivity anisotropic, with the longitudinal (along a root’s length) satu-rated conductivity of xylem (Kx ) being as much as seven orders of magnitude higherthan its radial counterpart (K0) (Sperry et al. 2002). The latter can be thought of as theharmonicmean of the low conductivities of inner and outermembranes, symplasm andxylem that comprise a root. Consequently, we treat the xylem hydraulic conductivityas a second-rank tensor,

K(ψ) = KsatKrel(ψ), Ksat =(K0 00 Kx

)

, Krel(ψ) = eαxψ. (6)

Dependence of the xylem’s relative conductivity Krel on matric potential ψ arisesdue to cavitation; in this context, the function Krel(ψ) is referred to as a “vulnerabilitycurve” (Sperry et al. 1998, 2002). Here, for the reasons described in the previoussection, we adopted the Gardner model of Krel(ψ) with the pore-size distributionparameter of the root xylem denoted by αx . Note that Gardner’s model (3) is a specialcase of the Weibull distribution Krel(ψ) = exp[−(−αxψ)c] used by Sperry et al.(1998) to represent vulnerability curves.

Finally, we use the steady-state Richards equation (1),

∇ · [K(ψ)∇ψ] + Kx∂

∂zKrel(ψ) = 0 (r, z) ∈ x , (7)

to describe flow in the root xylem.Plant physiology suggests that no water uptake occurs at a root’s end (Rand 1983;

Passioura 1988). Thus, following Arbogast et al. (1993) and others, we impose ano-flow boundary condition at the root cylinder’s bottom,

1 + ∂ψ

∂z= 0 z = B − L , 0 ≤ r < R. (8)

A given pressure (suction) ψ0 is prescribed at the soil surface, z= B, i.e. ψ(r, B)=ψ0.

2.4 Coupling conditions at root-soil interface

Let R− and R+ denote the limit r → R from the root and soil sides of the interface r =R, respectively. Conservation of momentum and mass imposes continuity conditionson, respectively, pressure head ψ and the radial components of Darcy’s flux qr ,

ψ(R−, z) = ψ(R+, z),

[

Kr (ψ)∂ψ

∂r

]

r=R−=

[

K (ψ)∂ψ

∂r

]

r=R+, B − L ≤ z ≤ B

(9)

where Kr is the radial component of the xylem conductivity tensorK given by (6).

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1652 G. Severino, D. M. Tartakovsky

These continuity conditions crystalize the difference between our “first-principle-based” approach and its empirical counterparts routinely used to model plant transpi-ration (e.g. Roose and Fowler 2004a, b; Green et al. 2006, and the references therein).Rather then enforcing the continuity of pressure across the root-soil interface r = R,the latter postulate the linear proportionality between the transpiration rate and thepressure (head) differenceψ(R+, z)−ψ(R−, z). We defer a further discussion of thisempirical approach to Sect. 4.

2.5 Integral transformation

We employ a modified Kirchhoff transformation,

�(r, z) ≡ eαz/2

ψ∫

−∞Krel(s)ds = α−1eα(ψ+z/2), α =

{αx (r, z) ∈ x

αs (r, z) ∈ s, (10)

to map the Richards equations (2) and (7) onto their linear counterparts

∇2� − α2s

4� = 0, (r, z) ∈ s (11a)

and

∇ · (Ksat∇�) − α2x

4Kx� = 0, (r, z) ∈ x . (11b)

The radial (qr ) and vertical (qz) components of the volumetric flux q = (qr , qz)defined in (1) are rewritten in terms of � as

qrKs

= −κre−αz/2 ∂�

∂r,

qzKs

= −κze−αz/2

(∂�

∂z+ α

2�

)

, (r, z) ∈ ,

(12a)

where

κr ={

κr = K0/Ks (r, z) ∈ x

1 (r, z) ∈ s, κz =

{κz = Kx/Ks (r, z) ∈ x

1 (r, z) ∈ s.

(12b)

The boundary conditions are transformed into

e−αs B/2[∂�

∂z+ αs

2�

]

z=B= − q

Ks, R < r < Reff ; (13)

�(r, 0) = α−1s , 0 ≤ r < Reff ; ∂�

∂r(r = Reff , z) = 0, 0 � z � B; (14)

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Flow at the soil-root interface 1653

and

[∂�

∂z+ αx

2�

]

z=B−L= 0, �(r, B) = α−1

x eαx (ψ0+B/2), 0 ≤ r ≤ R. (15)

The continuity conditions (9) at the root-soil interface� = {(r, z) : r = R, B−L ≤z ≤ B} take the form

(αx�−)αs = (αs�

+)αx , κre−αx z/2

(∂�

∂r

)−= −e−αs z/2

(∂�

∂r

)+, (16)

where �± ≡ �(R±, z) and (∂�/∂r)± ≡ ∂�/∂r(R±, z).

3 Perturbation solution

3.1 Dimensionless formulation

Both the identification of a small parameter and the subsequent perturbation analysisare facilitated by recasting the flow problem (11)–(16) in a dimensionless form. Weintroduce generic (as yet undefined) length scales �r , �z and �� to render dimensionlessthe coordinates and dependent variable,

r = r

�r, z = z

�z, � = �

��

. (17)

The Helmholtz equations (11) take a dimensionless form

(�z/�r )2K

r

∂

∂ r

(

r∂�

∂ r

)

+ ∂2�

∂ z2− α2

4� = 0, α = α �z; (r , z) ∈ (18)

where K = K0/Kx for (r , z) ∈ x and = 1 for (r , z) ∈ s . Here = {(r , z) :0 ≤ r < Reff , 0 ≤ z < B}, and the transformed root and soil domains are definedrespectively as x = {(r , z) : 0 ≤ r < R, B − L ≤ z < B} and s = /x , withR = R/�r , B = B/�z, L = L/�z and Reff = Reff/�r . The boundary and continuityconditions (13)–(16) are transformed into

��

�z

∂�

∂ z+ α��

2� =

{0 z = B − L, 0 < r < R

−q eαs B/2 z = B, R < r < Reff; (19)

�(r , 0) = 1

αs��

, 0 ≤ r < ∞; �(r , B) = eαx (ψ0+B/2)

αx��

, 0 ≤ r ≤ R; (20)

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1654 G. Severino, D. M. Tartakovsky

and

(αx���−)χ = αs���+, −κre−αx z/2

(∂�

∂ r

)−=e−αs z/2

(∂�

∂ r

)+, B − L ≤ z ≤ B;

(21)

where q = q/Ks , ψ0 = ψ0/�z , and χ = αs/αx . The dimensionless flux q =(qr , qz), with qr = qr/Ks and qz = qz/Ks , is given by

qr = −κr��

�re−αz/2 ∂�

∂ r, qz = −κz

��

�ze−αz/2

(∂�

∂ z+ α

2�

)

. (22)

3.2 Characteristic length scales and small parameter identification

For thick vadose zones (L � B) considered in this study, the flow is characterized byfour length scales: the root length (L) and radius (R), and the macroscopic capillarylengths of the root xylem (α−1

x ) and the ambient soil (α−1s ). We identify a small

parameter ε suitable for a perturbation analysis of (18)–(22) by relating �r , �z and ��

in (17) to these four characteristic length scales as follows. In the root’s absence, theflow (infiltration) would be vertical and described by the last two terms on the left-hand-side of (18). The root locally perturbs this background flow, introducing a radialdependence of the state variables in its vicinity. To account for this phenomenon, wedefine a small parameter in (18) as ε ≡ (�z/�r )

2 � 1, while requiring α ≡ α�z � ε.The same line of reasoning applied to the boundary conditions (19) and (20) suggestsorder relations ��/�z � ε and α�� � ε. In summary, we require the generic lengthscales �r , �z and �� to satisfy inequalities

(�z

�r

)2

≡ ε � 1, ε � α��, ε � ��

�z, ε � α�z . (23)

Selection of a set of the soil and root parameters L , R, αs and αx as the length scales�r , �z and �� is non-unique. It should be guided by site- and plant-specific values ofthese parameters and reflects a broad range of applicability of the perturbation solutionsderived below. According to its definition in (10), the parameter α takes the valuesof αx and αs in s and x , respectively. A possible choice of the length scales thatsatisfies the order relations (23) is

�z ≡ α−1s , �r ≡ R, �� ≡ L . (24)

A number of productive soils exhibit the macroscopic capillary length α−1s =

O(10−1 ÷ 10 cm) (Tartakovsky et al. 2003b, Fig. 1). Consider, as an example, maturemain roots of grapevine (Vitis viniferaL.)with the ratio of L/R = 42 cm /0.2 cmgrownin a soil with α−1

s = 10−1cm. For these parameters, (23) and (24) yield ε = 0.25,α�� ∼ O(102), ��/�z = 420 and α�z ∼ O(1). A large variety of chaparral plantsprovide another pertinent example. According to Hellmers et al. (1955), largest roots

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Flow at the soil-root interface 1655

of California scrub oak (Qifercuis duitnos Nutt.) exhibit R = 3.8 cm and L = 457cm. For the median range of capillary lengths, α−1

s = O(10−1 ÷ 1cm), this resultsin ε ∼ 6.25 × (10−6 ÷ 10−4), α�� ∼ O(102 ÷ 103), ��/�z ∼ O(102 ÷ 103)and α�z ∼ O(1). Another set of the length scales is provided by (24) in which thelast inequality is replaced by �� ≡ α−1

x . Thus the order relations (23) between thecharacteristic length scales account for a number of physical settings. In essence, oursolution is valid for roots whose size is larger than the macroscopic capillary lengthof a host soil.

3.3 Outer solution

We look for a solution of (18)–(21) in the form of an asymptotic expansion in thesmall parameter ε,

�(r , z) =∞∑

k=0

εk �k(r , z). (25)

Boundary-value problems for �k(r , z) are derived by substituting (25) into (18)–(21) and collecting the terms of k-th order in ε. In particular, the leading term in theexpansion (25) satisfies

∂2�0

∂ z2− α2

4�0 = 0, (26)

subject to the corresponding boundary conditions (19) and (20). This yields

�0(z) = e−αz/2

α ��

⎧⎪⎨

⎪⎩

eαx (ψ0+B) in x

1 − q (eαs z − 1) in s .

(27)

It follows from (22) and (27) that the zeroth-order approximation of the Darcianflux, q0 ≡ (qr0, qz0), has components

qr0 = 0, qz0 =

⎧⎪⎨

⎪⎩

0 in x

q in s .

(28)

The absence of derivatives with respect to r in the differential equation (26)—andthe resulting lack of radial dependence of its solution (27)—implies that the continuityconditions (21) on the interface r = R between s and x cannot be enforced. Thisindicates a singular nature of the asymptotic expansion (25), similar to that encounteredin perturbation analyses of viscous flow past a sphere (e.g. Cohen and Kundu 2004),free-surface flow to a well (Dagan 1968), and transport of kinetically sorbing solutesin porous media (e.g. Severino et al. 2006).

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1656 G. Severino, D. M. Tartakovsky

Higher-order terms in the expansion (25) can be computed in an identical manner.Combinedwith their leading-order counterpart (27), they describe vertical flow in boththe root (x ) and the ambient soil (s) except in the regions adjacent to the boundaryr = R separating the two. We proceed to derive a solution for this boundary layer.

3.4 Inner (boundary-layer) solution

In the region adjacent to the root-soil interface r = R, the pressureψ and its Kirchhofftransform � vary rapidly with the radius r . The derivatives with respect to r in (18)become large enough that even multiplied by ε � 1 they cannot be neglected. Toaccount for this behavior, we invoke the principle of the least degeneracy (Dyke 1975)to rescale the coordinates such that the leading-order term contains all the operatorcomponents that were neglected in the leading-order of the outer expansion. Withinthe boundary layer, we rescale the coordinates (r , z) as

z = ε1/4 z, r = ε−1/2

{r − R for r ≥ R

R − r for r ≤ R. (29)

Rewriting (18) in the (r , z) coordinates gives equations for � in the boundary layer(denoted below by �),

1

R + √ε r

∂

∂ r

[

(R + √ε r)

∂�

∂ r

]

+ √ε

∂2�

∂ z2− α2

s

4� = 0 in s (30a)

and

K0/Kx

R − √ε r

∂

∂ r

[

(R − √ε r)

∂�

∂ r

]

+ √ε

∂2�

∂ z2− α2

x

4� = 0 in x , (30b)

where s and x are the regions of the boundary layer outside and inside the root,respectively. Equations (30) are subject to the continuity conditions (21).

Following Verhulst (2005), we introduce an auxiliary function

φ(r , z) = �(r , z) −m∑

k=0

εk �k(z), (31)

where the last term represents the m-th order approximation of the outer expan-sion (25). This facilitates the subsequent matching of the inner and outer expansionsby ensuring zero overlap between the inner and outer solutions. It follows from (18)that

α2s

4�k − √

ε∂2�k

∂ z2≡ α2

s

4�k − ∂2�k

∂ z2= 0 for any k. (32)

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Flow at the soil-root interface 1657

Hence, substituting (31) into (30) yields

1

R + √ε r

∂

∂ r

[

(R + √ε r)

∂φ

∂ r

]

+ √ε

∂2φ

∂ z2− α2

s

4φ = 0 in s (33a)

and

K0/Kx

R − √ε r

∂

∂ r

[

(R − √ε r)

∂φ

∂ r

]

+ √ε

∂2φ

∂ z2− α2

x

4φ = 0 in x . (33b)

We look for a solution of (33) in the form of an asymptotic expansion

φ(r , z) =m∑

k=0

εk/2 φk(r , z) + O(εm+1/2). (34)

Substituting (34) into (33), using an expansion (R±√ε r)−1 = 1/R∓√

ε r/R2 +O(ε), and collecting the terms of order ε0, we obtain a leading-order equation

K∂2φ0

∂ r2− α2

4φ0 = 0 in s ∪ x . (35)

Its general solution is

φ0(r , z) = c(z) e−αr/2√K + b(z) eαr/2

√K in s ∪ x (36)

where c(z) and b(z) are “constants” of integration. For this expression to remain finiteas ε → 0, the definition of r in (29) requires that b(z) ≡ 0. Hence, it follows from (31)that

�0(r , z) = �0(z) + e−αr/2√K

{cx (z) in x

cs(z) in s. (37)

The constants of integration cx (z) and cs(z) are determined from the continuityconditions (21) at the root-soil interface r = 0, leading to

�0(r , z) = �0(z) + ξ(z)

�

exp

[α

2

(z

ε1/4− r√K

)] {Ks/

√K0Kx in x

1 in s(38)

where ξ(z) is a solution of the transcendental equation

[Ks ξ√K0Kx

+ eαx (ψ0+B−z/ε1/4)]χ

= ξ + e−αs z/ε1/4 − q(1 − e−αs z/ε1/4). (39)

123

1658 G. Severino, D. M. Tartakovsky

3.5 Matched asymptotic expansion and composite solution

According to Prandtl’s limit matching principle (e.g. Dyke 1975, Ch. 5.7) �u , auniformly valid (composite) zeroth-order approximation of �, is obtained by summingup the corresponding inner (�0) andouter (�0) solutions and subtracting their commonpart �com. Accounting for (38), this yields

�u = −�com + 2�0(z) + ξ

�

exp

[α

2

(z

ε1/4− r√K

)] {Ks/

√K0Kx in x

1 in s.

(40)

Since far away from the root (r → ∞) �u = �0, the common part �com = �0.Thus, (40) and (27) give

�u = 1

α ��

⎧⎪⎨

⎪⎩

eαx (ψ0+B−z/2) + Ksξ√K0Kx

e−αx�r (R−r)/(2√K) in x

(1 + q) e−αs z/2 − q eαs z/2 + ξ e−αs�r (r−R)/(2√K) in s

. (41)

This expression for the normalized Kirchhoff transform, αs�u exp(−αs z/2), is

plotted in Fig. 2 as a function of the dimensional radius αsr . It illustrates the boundary-layer behavior at the soil-root interface αs R = 2.0, for the parameter values reportedin Table 2. Note that the boundary layer is wider in the root than in the soil. This is in

0.0 1.0 2.0 3.0 4.0 5.0

0.0

0.5

1.0

1.5

2.0

norm

aliz

ed K

irch

hoff

pote

ntia

l

q = −0.01

q = −0.1

q = −0.5

q = −0.8

normalized radius, αsr

Fig. 2 Radial profiles of the normalized Kirchhoff potential αs�u exp(−αs z/2), for several values of thenormalized infiltration rate q = q/Ks . The root-soil interface is located at αs R = 2.0. The boundary layeraround the interface, within which the Kirchhoff potential changes with r , is asymmetric. Its dimensionlesswidth in the root is ∼ O(K√

ε), while that in the soil is ∼ O(√

ε)

123

Flow at the soil-root interface 1659

Table2

Parameter

values

used

inthesimulations

Parameter

value

αs

=10

.0Ks

=10

.0αx

=10

−3Kx

=0.1

K0

=10

−3R

=0.2

ψ0

=−1

03B

=10

3

Units

cm−1

cm/h

cm−1

cm/h

cm/h

cmcm

cm

Reference

Tartakovskyetal.

(200

3b)

Warrick

(200

3)Miller

(198

5)andPh

ilip

(196

8)

Sperry

etal.

(200

2)and

Frenschand

Steudle(198

9)

Sperry

etal.

(200

2)and

Frenschand

Steudle(198

9)

Doussan

etal.

(199

8)and

Mapfumoetal.

(199

4)

Allenetal.(19

98)

–

123

1660 G. Severino, D. M. Tartakovsky

0.0 1.0 2.0 3.0 4.0 5.0-12

-10

-8

-6

-4

-2

0

q = −0.001

q = −0.01

q = −0.1

q = −0.5

normalized radius, αsr

norm

aliz

edpr

essu

re,α

sψ

Fig. 3 Radial profiles of normalized pressure αsψ , for several values of normalized infiltration rate q =q/Ks . The root-soil interface is located at αs R = 2.0. Pressure ψ is maximum at the root-soil interface,from which it decays faster in the root than in the soil. Changes in infiltration rate q affect the pressure inthe root more than in the soil

accordance with (38), which predicts that the widths of the boundary layer in the rootand soil are O(K√

ε) and O(√

ε), respectively.Pressure distribution ψ(r , z) in the root-soil continuum is computed from (10)

as α ψ = −α z/2 + ln(α ���u). Its graphical representation in Fig. 3 reveals thatthe pressure ψ reaches its maximum value at the root-soil interface r = R, fromwhich it decreases linearly inside the root (r < R) and monotonically outside theroot (r > R). Away from the root (r � R) the pressure reaches its asymptoticvalue of αsψ∞ = −αs z + ln

[1 − q (eαs z − 1)

]. The pressure distribution predicted

with (41), and depicted in Fig. 3, supports plant physiology studies [see e.g. Rand(1983), Passioura (1988)], which found that water suction (negative pressure) alongthe xylem is at least an order of magnitude higher than that in both the root-soilinterface and the ambient soil.

The uniformly valid expression for the Darcian flux qu = (qur , quz ) is obtained bysubstituting (41) into (22),

qur (r, z) = δ

2ξ exp

[

− δ α

2√K (r − R)

]

, quz (r, z) = qz0− 2 δ√K

(

1+ ξ ′

α ξ

)

qur (r, z),

(42)

where ξ ′ is the derivative of ξ(z). The vertical component quz approaches (exponen-tially) its far-field value of qz0 as r becomes large. This solution reflects the fact thatfar away from the root-soil interface the flux qu is vertical (Fig. 4). The radial com-

123

Flow at the soil-root interface 1661

q = −0.01

q = −0.1

q = −0.5

q = −0.8

0.0

0.0

-0.2

-0.4

-0.6

-0.8

-1.00.1 0.2 0.3 0.4 0.5

normalized radius, αsr

norm

aliz

edve

rtic

alco

mpo

nent

offlu

x,q z

=q z

/Ks

Fig. 4 Radial profiles of normalized vertical flux qz = qz/Ks , for several values of normalized infiltrationrate q = q/Ks . The root-soil interface is located at αs R = 2. Outside the boundary layer, the vertical fluxin the soil quickly reaches its far-field value qz0 as r increases

ponent qur is non-zero only in the zone adjacent to the root-soil interface, decreasingexponentially with distance r (Fig. 5).

4 Results and discussion

The closed-form analytical solutions developed above enable us to derive from “firstprinciples” two fundamental quantities of interest, plant’s transpiration rate and root’scapture zone. Unless specified otherwise, all the figures presented in this section corre-spond to the parameter values summarized in Table 2. They give rise to dimensionlessparameters αs B = −αsψ0 = 104, Ks/

√K0Kx = 103, αs z = 9800, αs R = 2, and

K = 10−2.

4.1 Transpiration rate

For a root with circular cross-section A = πR2, the Darcian flux in the root zonequz (r) and the plant transpiration rate Qt are related by Qt = 2π

∫ R0 |quz (r)| r dr . The

corresponding average transpiration flux J � is defined as J � = Qt/(πR2). Using thedimensional form of quz (r) in (42), we obtain

J � =8 Ks

(K0/Kx

αx R

)2 (

ξ+ ξ ′

αx

) [αx R

2

√Kx

K0+exp

(

−αx R

2

√Kx

K0

)

−1

]

. (43)

123

1662 G. Severino, D. M. Tartakovsky

normalized radius, αsr

0.0 1.0 2.0 3.0 4.0 5.00.000

0.001

0.002

0.003

0.004

0.005

q = −0.01

−0.8

−0.5

−0.1

norm

aliz

edra

dial

com

pone

ntof

flux,

q r=

q r/K

s

q = −0.01

q = −0.1

q = −0.5

q = −0.8

Fig. 5 Normalized radial flux qr = qr /Ks as a function of normalized radial distance αsr , for severalvalues of normalized infiltration rate q = q/Ks . The root-soil interface is located at αs R = 2. The radialflux is non-zero only in the boundary layer around the root-soil interface. Significant anisotropy of the rootconductivity (K � 1) is responsible for the asymmetry of the boundary layer within which qr decays tozero (much faster in the root than in the soil)

To the best of our knowledge, (43) is the first closed-form analytical expressionthat relates plant transpiration to the soil (Ks and αs) and root (K0, Kx , αx , and R)properties. It provides physical insight into the coupled nonlinear processes involvedin root water uptake and, more generally, plant transpiration.

For a given set of hydraulic properties of the root xylem (K0, Kx and αx ) andthe soil (Ks and αs), the average transpiration flux J � (Fig. 6a) decreases, and thetranspiration rate Qt (Fig. 6b) increases, with the root radius R. While the former isin the dimensionless form that represents a wide range parameter combinations, thelatter corresponds to the hydraulic properties specified in the beginning of this section.The values of Qt in Fig. 6b are in line with the measurements (e.g. Steppe et al. 2005).The mean transpiration flux J � is related to the pressure ψ at the center of the root.In particular, for z = B the value ψ� of the pressure determining the transpirationflux (43) is

ψ� = α−1x ln

[Ks ξ(B)√K0Kx

e−R + eαxψ0

]

. (44)

These estimates of transpiration flux J � and root suction ψ� come with a caveatassociated with the steady-state nature of our model (see the discussion in Sect. 2.2).While vadose zoneprocesses are seldomstationary, the steady-state assumption is validfor precipitation or irrigation events leading to prolonged infiltration (e.g. Severinoand Indelman 2004). As discussed in Sect. 2.2, steady-state solutions are buildingblocks in transient models that represent plant transpiration by a sequence of steadystates. Moreover, in practice, plant transpiration rate is estimated by multiplying themaximum (i.e. steady-state) transpiration with a crop coefficient (Allen et al. 1998).

123

Flow at the soil-root interface 1663

0 0.001 0.01 0.1 1 10 100 1000 10000

0.0

0.2

0.4

0.6

0.8

1.0

normalized root radius, R

norm

aliz

edav

erag

etr

ansp

irat

ion

flux,

J�/(

Ksq� r

)

(a)

0.01 0.1 1 10

0.01

0.1

10−3

10−4

10−5

(b)

root radius, R [cm]

tota

ltr

ansp

irat

ion

rate

,Qt

[L/h

]

Fig. 6 a Average transpiration flux J �, normalized with Ksq�r = Ks (K0/Kx )

2(ξ + ξ ′/αx ), as a functionof the scaled root radiusR. Our model predicts that this relationship is a universal feature of water uptakeby a single root, which is independent of the soil and root hydraulic properties. b Dependence of thetranspiration rate Qt on the root radius R for the parameter values listed in the Table 2

4.2 Capture zone

The soil region affected by a root’s water uptake is referred to as a plant capture zone(PCZ). Estimating its size is essential for agriculture and phytoremediation, a process

123

1664 G. Severino, D. M. Tartakovsky

0 4 8 12 16

0

10

20

30

40

50

ρ/ρc = 1

0.8

0.6

0.4

0.2

δρ = 1.0

normalized radial coordinate, ρ = αs(r − R)

norm

aliz

edtr

ajec

tory

,α

sZ

(ρ;δ

ρ)

Fig. 7 Scaled trajectories αs Z(r; δρ) of particles released on the soil surface at a relative distance δρ fromthe root. The solid (red) line δρ = 1.0 delineates the plant capture zone (PCZ). Particles released on thesoil surface at distance r ≤ R�

p (or δρ ≤ 1.0) are captured by the root

in which plants are used to extract pollutants from contaminated soils. To delineate thePCZ, we compute a trajectory Z = Z(r; Rp) of a particle released on the soil surface(Z = B) at a distance r = Rp by solving numerically an ordinary differential equation

dZ

dr= quz (r, Z)

qur (r, Z). (45)

A trajectory bounding the PCZ originates on the soil surface at the critical distanceR�p, which is defined as a solution of Z(0; R�

p) = L where L is the root’s length.Let ρ = αs(r − R) denote the dimensionless radial coordinate that varies between

the root (r = R) and a point r = Rp > R on the soil surface at which a particle hasbeen released. Let us define a dimensionless critical distance ρ� = αs(R�

p − R) andintroduce a maximum distance ratio δρ = (Rp − R)/(R�

p − R). Trajectories Z(ρ; δρ)

with a label δρ ≤ 1 belong to the PCZ (Fig. 7). In other words, any particle releasedon the soil surface at distance r ≤ R�

p will be captured by the root, while particlesreleased at distance r > R�

p will not.Figure 8 reveals that the dimensionless critical distance ρ� decreases monotoni-

cally with the normalized infiltration rate q/Ks . This is to be expected, since smallerinfiltration values cause the root to draw water from larger volumes of the ambientsoil. For the same reason, ρ� increases with the dimensionless root length αs L .

Combined with the definition of dimensionless critical distance ρ�, Fig. 8 suggeststhat for a given infiltration rate q a root in a soil with larger saturated conductivity Ks

and/or the Gardner parameter αs would have a larger capture zone (i.e. larger ρ� or

123

Flow at the soil-root interface 1665

0.001 0.01 0.1 10

5

10

15

20

αsL = 10

20

30

4050

normalized infiltration flux, q/Ks

dim

ensi

onle

sscr

itic

aldi

stan

ce,ρ

�

Fig. 8 Dependence of the critical distance of the plant capture zone, ρ�, on infiltrating rate q/Ks , forseveral values of scaled root length αs L . The critical distance defines the maximum radial extent of thecapture zone. The monotonic decrease of ρ� with q/Ks indicates that smaller infiltration values cause theroot to draw water from larger volumes of the ambient soil. For the same reason, ρ� increases with thedimensionless root length αs L

R�p). Larger values of Ks result in larger values of vertical flux qz , thus enhancing the

vertical infiltration and hence reducing the amount of water available for root wateruptake. Large values of αs have a similar effect. They represent soils with reduced soil-water retention, which are conducive to gravity-driven (vertical) flows (Philip 1968),once again reducing the amount of water available for root water uptake and causingthe root to “interrogate” large volumes of the ambient soil. Finally, Fig. 8 suggeststhat, for given soil and root parameters, roots with larger lengths L have larger ρ� andhence capture zones.

4.3 Comparison with empirical models of plant transpiration

Our root-scale model of water uptake is based on first principles, i.e. employs thegenerally accepted Richards equation to describe water flow in partially saturatedporous media (both in a root and the ambient soil) and makes no assumptions aboutthe kinematic structure of flow in a root-soil continuum. That is in contrast with theexisting root-scale (Type I) and mesoscale (Type II) models (Green et al. 2006; Raats2007): the former impose a priori constraints on the kinematic structure of flow, suchas the assumptions (ii)–(iv) discussed in the Introduction; and the latter are entirelyempirical. The classification of root water uptake models into Types I and II is due toGreen et al. (2006).

Our analytical solution (see Sect. 3.4) provides a theoretical justification for severalassumptions that underpin the existing Type I models. Specifically, it demonstrates

123

1666 G. Severino, D. M. Tartakovsky

that flow is strictly horizontal (perpendicular to the root) in the boundary layer on bothside of the soil-root interface. This provides a theoretical justification for empiricalinfinite-soil-cylinder mesoscopic models (Raats 2007, and the references therein) .They assume that flow is horizontal and driven by the difference in matric potentialsψ0 and ψ1 imposed at the surfaces of the soil shell R < r < r1, where r1 is anarbitrarily assigned external radius of the soil cylinder. Our solution suggests that theradius r1 of such soil cylinders is given by the boundary layer thickness ε. The lattervaries with the root geometry and/or hydraulic properties of the root and the ambientsoil, as discussed in Sect. 3.2.

5 Summary and concluding remarks

We derived, from first principles, a closed-form analytical solution for plant transpi-ration, i.e. for flow of water from the partially saturated ambient soil to a plant’s root.The underlying model consists of two three-dimensional Richards’ equations that arecoupled at the root-soil interface. The root is conceptualized as a cylinder of lengthL and radius R. The system behavior is controlled by vertical and horizontal lengthscales �z and �r . For a wide range of natural conditions, ε = (�z/�r )

2 � 1 and servesas a perturbation parameter.

A matched asymptotic expansion technique was used to derive approximate solu-tions for transpiration rate and the size of a plant capture zone (PCZ). To the best ofour knowledge, this is the first closed-form analytical expression that relates a plant’stranspiration rate to the soil and root hydraulic properties by relying on first principles,rather than phenomenology. As such, it sheds new light on plant transpiration, one ofthe least-understood components of the hydrological cycle.

Our analysis leads to the following major conclusions.

– Away from the root-soil interface the water flux q is vertical. The radial componentof q is non-zero only within the boundary layer adjacent to the root-soil interface.This provides a theoretical justification for the currently used empirical “infinitesoil cylinder” mesoscopic models.

– The radial component of q decays exponentially with the radial distance from thesoil-root interface as consequence of the rapid variation due to the boundary-layertransitional effect.

– The PCZ size increases as the infiltration rate decreases relative to the radial fluxat the soil-root interface.

Several assumptions underpin the presented model (see Sect. 2). Some of them canbe relaxed or altogether eliminated within our analytical framework. � The assump-tion of soil homogeneity was used to treat soil hydraulic properties as constants. Itcan be relaxed by assuming that a soil is statistically homogeneous (i.e. its parametershave constant ensemble means and variances) and employing stochastic homoge-nization (Tartakovsky et al. 2003a, b). � The steady-state assumption can be elim-inated by adopting an exponential pressure (ψ) vs. saturation (θ ) constitutive law,θ ∼ exp(αsψ). This would allow one to account for transient flow regimes, whileretaining the ability to linearize the Richards equation by deploying the Kirchhofftransformation (Tartakovsky et al. 2004). � Finally, our model represents a root system

123

Flow at the soil-root interface 1667

with a single cylinder. It can serve as a building block in more realistic models (Rooseand Fowler 2004a), which represent root networks as branching systems of cylinders.

These generalizations are the focus of our ongoing studies.

Acknowledgments This research was supported in part by the National Science Foundation award EAR-1246315 and by theComputationalMathematics Programof theAir ForceOffice of ScientificResearch. Thefirst author acknowledges support from “Programma di scambi internazionali per mobilitá di breve durata”(NaplesUniversity, Italy), “OECDCooperativeResearch Programme:Biological ResourceManagement forSustainable Agricultural Systems” (Contract No. JA00073336), and PRIN project “I paesaggi tradizionalidell’agricoltura italiana: definizione di un modello interpretativo multidisciplinare e multiscala finalizzatoalla pianificazione e alla gestione” (Contract No. 2010LE4NBM_007). The first author thanks Prof. GerardoToraldo (Naples University) for promoting his visit to University of California, San Diego; and Dr. PengWang for his kind hospitality.

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