WIMOVAC Soil Processes

Soil-Plant Water Relations

Models of water uptake by plant roots generally have one of two purposes. Either they produce estimates of transpirational water loss for water budget models or they provide estimates for predicting plant water stress {Campbell, 1991 #1958}. Wimovac uses a simple computer algorithm to simulate multiple layer soil water flow, plant water uptake and plant-water relations for a given soil-plant-atmosphere system. This is based upon the analytical plant water uptake models derived by Campbell (1991) but modified here to use wimovac’s own soil evaporation and canopy transpiration routines in preference to the simpler formulations proposed by Campbell. Detailed reviews and analyses of plant water uptake have been published {Molz, 1981 #1996; Hillel, 1980 #1995} and form the background of this model module.

In the development of a theory to describe plant water uptake and loss, it has been useful to employ electrical analogs of the system {Gardner, 1960 #1994; Cowan, 1965 #1992; Van den Honert, 1948 #1993}. Figure 33 shows the electrical analog used by Cowan (1965) of a plant withdrawing water from a uniformly rooted soil layer (only resistances in the liquid phase are shown).

Potentials and resistances in the soil-plant system. E is the transpiration rate, y_L is the leaf water potential, y_xL is the xylem/leaf water potential , y_xr is the xylem/root water potential, y_r root water potential, y_s is the soil water potential, R_L is the leaf resistance, R_x is the xylem resistance, R_r is the root resistance, R_I is the root interfacial resistance and R_s is the soil resistance. Adapted from Campbell (1991)

Cowan (1965) assumes that transpiration rate (E) is regulated mainly by vapour phase driving forces and resistances. Plant water potentials are therefore primarily the result of an imposed rate of transpiration, from the leaf/canopy transpiration module, and stomatal conductance rather than the converse situation in which plant water potentials are the direct determinant of E. The major resistances in Cowan’s model are outlined in Figure 33 and consist of resistances due to the soil (R_s), soil interface (R_I), root endodermis (R_r), stem xylem (R_x) and leaf (R_L). For most species grown in well watered, medium textured soil, the important resistances are due to the root endodermis (R_r) and the leaf (R_L). In rough numbers a representative mid-day transpiration rate (E) of 2.3^.10^-4 Kg m^-2 s^-1, in a mature plant, produces leaf water potentials of about -1500 kPa in exposed leaves and -1000 kPa in darkened leaves and from this observation Cowan was able to parameterise Figure 33 such that when y_s=0, y_L=-1500 kPa and y_xL=-1000 kPa. Cowan (1965) then assumed that the xylem and soil resistances were negligible and could calculate R_r and R_L using Ohm’s law. Axial resistance in the root, where important was assumed to be included in the R_r value. The root resistance calculated in this manner is the total resistance of the root system. If the length of root is known the resistance per unit length can be calculated and typical values of around 2.5^.10¹⁰ m³ kg^-1 s^-1 per unit length have observed {Bristow, 1984 #1997}.

Following the analysis of Cowan (1965) it is then possible to write a differential equation for the uptake of water by a single root as:

where q_w is the flux of water (kg s^-1 ), k is the soil hydraulic conductivity (kg s m^-3), y is the matrix potential (k Pa) and r is the radial distance from the root axis (m). The area for water flow is A_w=2prl where l is the length of root. Hydraulic conductivity can then be related to water potential using the term

where k_s is saturated conductivity (kg s m^-3), y_e is the air entry potential (J kg^-1) and n is a constant that depends on soil texture and ranges typically from 2 to 3.5 {Campbell, 1974 #1998}. It is then possible to combine and integrate Equation 132 and Equation 133 from the bulk soil at r_s, where water potential is y_s, to the root surface (r_r) where the potential is y_r, to give an expression (Equation 134) that may be related to measurable values.

where L is the rooting density and Dz is rooting depth. Using this and the expression of Gardner (1960) {Gardner, 1960 #1994}for r_s it is possible to calculate the soil resistance term as:

Since B depends on root depth and root density, soil resistance in this model is therefore a function of these variables. It will also vary with soil water potential and to some degree with transpiration rate (E).

Few natural rooting systems would fit Cowan’s initial assumption of a uniform rooting density and this model in itself would be of little use. However Campbell (1991) has been able to extend this simple model to include variable rooting density conditions in which the soil profile is assumed to be made up of a series of zones or layers each of which has a characteristic root density, soil water potential and associated resistances and it is this model that has been incorporated into wimovac.

In Campbell’s (1991) model water is assumed to move from the soil, through the plant, to the evaporating surfaces in the sub-stomatal cavities in response to gradients in water potential. Only matric and gravitational potential are explicitly accounted for in the derivation, and resistances to flow in the liquid phase are assumed to be mainly in the soil, the root endodermis and the leaf. These resistances are assumed constant and known during the period of a single simulation time step (15 minutes). Interfacial and xylem resistance could have been included but have been assumed to be negligible for the purpose of simplification in the model.

The water uptake from each soil zone in Campbell’s model is described by the equations previously used by Cowan (1965) to describe a single layer since the rooting density within a single zone can be assumed to be constant. The total uptake of water from the soil is the sum of the uptake from each zone (Equation 136) where y_si is the soil matric potential of zone i, g is the gravitational constant (9.8 m s^-2), z_i is the depth of layer i, y_x is the xylem water potential, R_si is the soil resistance in zone i and R_ri is the root resistance calculated from Equation 137 in which L_i is the rooting density of soil zone i. Equation 137 expresses the underlying assumption of the Campbell model that R_ri is directly proportional to the total root resistance and inversely proportional to the fraction of the root system that is in that layer. It is possible to then calculate the crown water potential of the plant (y_x) using Equation 138 and to use this to calculate the leaf water potential y_L using Equation 139. y_L must then be solved simultaneously with another equation that expresses the effect of y_L on E through stomatal conductance (Equation 43, Equation 44) and this is achieved using an iterative process.

Figure 34 shows typical diurnal output from the soil-plant water potential model for a an initially well watered clay-loam soil during a 10 day dry down period and the values generated for both leaf and soil layer water potentials are in keeping with a number of field observations {Turner, 1983 #2000; Campbell, 1976 #1999}. The diurnal patterns of leaf and soil-water potential variation are important because they drive the stomatal closure model (Equation 43) which in turn strongly influences canopy transpiration (E), assimilation (A) and energy budget. The short time step that the model operates with (between 15 minutes and 2 hours) allows this model to express midday stress on stomatal conductance with the consequent depression in midday transpiration and assimilation typically observed in field trials {Campbell, 1976 #1999}.

Figure 34iii shows that the weighted mean soil-water potential for all soil layers is strongly influenced by the densely rooted layers at the top of the soil profile. The fact that mean water content (Figure 34iv) increases during the night means that the water potential in these layers increases overnight as well. Campbell’s model does not artificially restrict movement of water from the roots to the soil and assumes that that resistance to water flow is the same for water flowing out of roots as it is for water flowing in. This assumption is supported by work by Baker and van Bavel (1986){Baker, 1986 #2001}.

Soil-plant-water relations. i) Predicted decline in average leaf water potential during a 10 day dry down period (British, mid-summer) starting from a well watered condition. F_canopy=3. ii) Increased stomatal closure resulting from lowered leaf water potential iii) Decline in soil water potential of 5 uppermost soil layers (total soil depth represented by layers 1m). iv) Decline in volumetric water content of the 5 uppermost soil layers. All parameter values as stated in Campbell (1976) unless otherwise stated in appendices I & II.

In addition to the relatively complex soil-plant model of Campbell (1976) wimovac also contains a much simpler model of volumetric soil water content dynamics as proposed by Johnson (1993). {Johnson, 1993 #1919}

In his approach Johnson (1993) assumes a simple 1 layer soil system in which there is no evaporation from the soil, runoff or drainage. The basis of the model is to calculate the potential daily transpiration rate (the transpiration rate that occurs when there is no resistance to water vapour transfer between the soil and the atmosphere) using either the Penman-Monteith or the Priestly-Taylor formulations (Equation 146 and Equation 147 respectively) and then to assume that the actual daily transpiration is related to soil water content (q_soil). For values of q_soil greater than a particular critical value, q¹, (which you can define), transpiration occurs at the potential rate, and for values of q_soil less than another critical value, q² (the wilting point), there is no transpiration. Intermediate values of q_soil lead to transpiration rates linearly distributed between zero and the potential rate. Any soil type can therefore be described in this model using three simple parameters, i). the field capacity (q^*). ii). the critical value (q¹). iii). and the wilting point (q²). The default values for these parameters are included in wimovac in the form of a small soil database that includes entries for clay, clay-loam, sandy loam and user defined soil types. The daily soil water balance is given by Equation 141 where the subscripts i and i+1 refer to days i and i+1, r_w is the density of water and d_s is the soil depth. Simplicity of parameterisation and design are the main advantages to this approach but the direct, albeit empirical, manner in which the rate of soil water loss is linked to soil water content provides a reasonable simulation of the observed reduction in water loss rate associated with soil dry down evident in a number of field observations (Campbell, 1976).

Equations from Campbell (1974) and Cowan (1965){Campbell, 1974 #1998; Cowan, 1965 #1992}.

 Soil surface evaporation

The evaporation of water directly from the soil is often an important component of effective water loss for the canopy, but is strongly dependent on soil wetness and on plant cover {Jones, 1992 #1954}. Several reports indicate that, even with wet soil, soil evaporation is only about 5% of the total when the leaf area index reaches 4 or more, for crops as diverse as wheat and Pinus radiata {Brun, 1972 #1894; Denmead, 1969 #1893}{Denmead, 1969 #1893}{Brun, 1972 #1894}. When the leaf area index is 2 or less, however, wet soil evaporation can be as much as half the total. Ritchie (1972){Ritchie, 1972 #1990} has suggested that after rain about 10mm of soil evaporation can occur at a high rate (i.e. the soil surface resistance is close to zero). As the soil dries, the surface resistance increases and soil evaporation decreases significantly. With a dry soil surface, soil evaporation is relatively unimportant, and evaporation is solely transpiration from plant leaves and this is approximately proportional to leaf area index {Jones, 1992 #1954}.

Wimovac incorporates two alternative models of soil evaporation. i). The first is commonly called the Priestly/Taylor approach and calculates a potential evaporation rate by assuming that there is no resistance to water flux from the soil surface. This is equivalent to the Penman-Monteith equation for potential transpiration if the diffusion term is 26% of the radiation value. There appears to be little theoretical basis for this simplification, but it is widely used by many modellers and is included here for comparison. ii). The alternative soil evaporation model is based upon the Penman-Monteith formulation which is similar in structure to that used for the calculation of canopy transpiration described previously. This formulation is generally considered more realistic and uses a soil boundary layer conductance term to give actual evaporation rather than the potential evaporation produced by the Priestly/Taylor method.

Both of the soil evaporation models calculate the total incident solar radiation on the soil surface and use this to calculate the net radiation balance (F _N,soil) of the soil by subtracting the longwave radiation re-emitted, from the soil, from the total incident radiation (Equation 145). For the sunlit/shaded leaf canopy model, total incident solar radiation for the soil is assumed to be equal to the leaf area weighted average light intensity incident on sunlit and shaded leaves, and for the multiple layer canopy it is assumed to be equal to the leaf area weighted average light intensity of sunlit and shaded leaves in the bottom canopy layer. The radiation energy loss due to re-emission from the soil is assumed to be given by Stefans law reworked for an emitter and air at a similar temperature (Equation 143). The soil boundary layer conductance g_a,soil is calculated using an empirical expression (Equation 142) containing a term for the boundary layer thickness () where (u_soil) is the wind speed at the soil surface and (S_size) is the average soil particle size and a diffusion component .

Both soil evaporation models show the expected reduction in soil evaporation rate associated with larger canopy leaf areas but neither adequately takes into account the increase in effective soil resistance that occurs during top layer soil drying. Consequently both models may be prone to overestimate soil evaporation under dry soil conditions. The Penman-Monteith boundary layer conductance (Equation 142) term could be modified to include a soil resistance factor produced by the soil water potential module but this is beyond the scope of current work and may represent an unnecessary complication.

Equations from Johnson (1993).{Johnson, 1993 #1919}.

Heat flux & temperature

Models of soil heat flux and associated temperature estimates typically make two major contributions to soil and crop models. Firstly they describes energy partitioning at the soil surface which directly and indirectly affects plant growth and development, and secondly they describe the soil temperature distribution which affects many of the biological and physical processes associated with the soil.

Wimovac uses a one dimensional multiple layer model of soil temperature and heat fluxes identical in structure to that used for the soil water potential calculation module. There are four methods of generating soil temperature and heat flux information included in wimovac and these may be selected by using the appropriate switch in the soil microclimate-temperature section of the model parameter database. These options reflect a series of trade off’s between ease of parameterisation and mechanistic detail. i). The first option simply sets all soil layers to a given fixed temperature which may be specified at the start of the simulation run in the model parameter database. ii). The second option allows each soil layer to have a different temperature fixed at the beginning of the simulation. iii). The third option sets all soil layer temperatures equal to the ambient air temperature which changes dynamically with time of day and day of year. There is little theoretical or experimental justification for any of these options but they represent simple methods of specifying soil temperature and are useful for debugging other complex model components which require an input of soil properties. iv). The final option is by far the most mechanistically rich approach and attempts to use the physically correct model proposed by Horton and Chung (1991).

Soil is represented in this model as a complex 3 phase system in which there are solid particulates, a liquid phase water based flux and an associated vapour flux arising from various physical and biological processes. Each type of soil represents a potentially different balance of these components and a successful model of soil heat flux and temperature must take these into account. The model proposed by Horton and Chung (1991), and used in wimovac, assumes that {Horton, 1991 #1959} there are three mechanisms, radiation, convection, and conduction responsible, simultaneously, for the transfer of heat in soil. Radiative energy transfer includes incoming direct and diffuse shortwave solar radiation, longwave sky radiation to the soil surface, and longwave radiation re-emitted outward from the soil surface. Convective energy transfer in a porous media such as soil is associated with a net flux of fluids. A number of authors {Jackson, 1960 #2003; Wierenga, 1969 #2004} have studied the convective energy transfer in soils associated with vapour and liquid fluxes in the soil and although convection may be a major portion of heat flow during periods of large moisture flux, such as during rainfall or irrigation, the mechanism most responsible for subsurface heat transfer is conduction {Horton, 1991 #1959}.

Philip and De Vries (1957) {Philip, 1957 #2008}presented a theory to describe coupled water and heat flow in soil. Van Bavel and Hillel (1975, 1976){Van Bavel, 1976 #2010}{Van Bavel, 1975 #2009}, Milly (1982) {Milly, 1982 #2011} and Bristow et al., (1986) {Bristow, 1986 #2012}expanded and/or used this theory to calculate heat and water flows in soil and were able to derive a description (Equation 148) for conductive heat transfer for a vertical one dimensional soil system.

where C_soil is the volumetric heat capacity (J ° C^-1 m^-3), T_a is the temperature (° C), t is the time(s), z is the depth (positive downward, m), and l_soil is the thermal conductivity (W m^-1 ° C^-1). In the presence of a temperature gradient in a moist soil, heat transfer takes place by convection, in addition to conductance, but by making l_soil the apparent rather than the real thermal conductivity Horton and Chung (1991) avoid having to include this term explicitly. In a moist soil l_soil depends on the soil water content which can vary with both depth and time.

The movement of water in the one dimensional soil profile is described by Equation 149.

where F is the specific water capacity, , and where q_soil is the volumetric soil water content, h_soil is the soil water pressure head, K_soil is the hydraulic conductivity, S is the source or sink term and z_soil is the vertical distance downward from the soil surface.

Horton and Chung (1991) showed that Equation 148 and Equation 149 can then be solved simultaneously provided that the initial and boundary conditions are known.

The initial conditions in the wimovac version of this model are specified in the model parameter database and consist of initial soil layer temperature information and various soil physical properties (see appendix II). Boundary conditions are calculated on the basis of an energy balance method at the soil surface which is used to determine thermal and hydraulic upper boundary properties. Soil evaporation is calculated in the soil surface evaporation module (Equation 142 to Equation 147) and is used in conjunction with incident and re-emitted solar radiation in order to calculate the latent and sensible heat fluxes at the surface. Soil surface evaporative flux and soil surface temperature are not independent quantities and so are estimated in the model in an iterative fashion. The humidity at the soil surface is calculated from Equation 150, where HO_soil is the saturated humidity at the soil surface (kg m^-3) and h_soil is the soil water pressure head.

The saturated humidity, HO_soil, at the soil surface is calculated using Equation 151 derived by Murray (1967) {Murray, 1967 #1879} and the soil heat flux density, G_soil, at the soil surface may then be described by Equation 152 where l_soil is the thermal conductivity of the soil surface (W m^-1 °C^-1). Net radiation, soil evaporation, soil surface humidity and G_soil are all functions of the unknown soil surface temperature and before the surface temperature can be calculated an approximation for G_soil must be used (Equation 153) , this is provided by Chung and Horton (1987){Chung, 1987 #2013} in which T_s is the surface temperature for the present time step, T₁ is the surface temperature from the previous time step, T₂ is the soil temperature from the previous time step for the node at vertical position 2, Dz is the vertical spatial increment, C is the volumetric heat capacity for the soil surface layer and Dt is the time step increment.

Equation 153 approximates the soil heat flux density by adding a term that estimates soil heat flux at a depth of and a term that estimates the change in heat stored in the soil above . The value of T_s is solved numerically for each time step using a bisector root-finding algorithm {James, 1977 #2014}. The predicted soil surface temperature, T_s from the energy balance partitioning is used as the upper-boundary condition. The model algorithm then uses an implicit, finite difference method (Equation 154) to solve the soil heat and water flow equations.

Soil temperature profiles. i). Predicted soil temperature profile for J_day=190 at 0400hr and 1600hr indicating the change in temperature in the upper layers. Note that the upper levels are more directly connected to air temperature compared to lower layers which are relatively more stable. ii).Predicted soil temperature surface for J_day=100 to J_day=101. Total depth of soil profile=0.5m, F_canopy=3. Parameters values as stated in Horton and Chung (1991) unless otherwise specified in appendices I & II.

Soil parameters required by the model as inputs include soil surface emissivity, initial soil surface albedo, soil thermal conductivity, soil volumetric heat capacity, soil water capacity, and hydraulic conductivity, all as functions of water content, roughness length, and the soil-water characteristic curve. A few general inputs are also required to run the module and these are obtained from the appropriate routines in the macroclimate module and include the time of solar noon, day length, length of simulation, D z, and D l and various weather inputs such as rainfall. Soil water characteristics, hydraulic conductivity and specific water capacity are described by empirical equations presented by Van Genuchten (1980) as given by Equation 155 in which K_s is the saturated hydraulic conductivity, h is the absolute value of the pressure head and a ₁ and n are non linear regression parameters describing the shape of the soil water characteristic curve. Soil surface emissivity, e , (Equation 157) follows that used by Van Bavel and Hillel (1976).{Van Bavel, 1975 #2009} The soil volumetric heat capacity (J °C^-1 m³) is determined following De Vries (1963) {De Vries, 1963 #2015}and is given by Equation 158, where q _s is the input saturated volumetric water content of a specified soil. Soil thermal conductivity is assumed to be related to water content by the empirical expression (Equation 159). Soil surface albedo (a_soil) may be obtained as a fixed value from the model parameter database or determined according to Van Bavel and Hillel (1976) as indicated in Equation 160.

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