Soil heat and water flow with a partial surface mulch

Transcription

1 Agronomy Publcatons Agronomy Sol heat and water flow wth a partal surface mulch Sang-Ok Chung owa State Unversty Robert Horton Jr. owa State Unversty, rhorton@astate.edu Follow ths and addtonal works at: Part of the Agronomy and Crop Scences Commons, Fresh Water Studes Commons, Plant Bology Commons, and the Sol Scence Commons The complete bblographc nformaton for ths tem can be found at agron_pubs/318. For nformaton on how to cte ths tem, please vst howtocte.html. Ths Artcle s brought to you for free and open access by the Agronomy at owa State Unversty Dgtal Repostory. t has been accepted for ncluson n Agronomy Publcatons by an authorzed admnstrator of owa State Unversty Dgtal Repostory. For more nformaton, please contact dgrep@astate.edu.

2 Sol heat and water flow wth a partal surface mulch Abstract A computer model usng the alternatng drecton mplct (AD) fnte dfference method to study twodmensonal coupled sol heat and water flow wth a partal surface mulch cover s developed. A new, smplfed computatonal procedure, whch has only trdagonal matrx problems, for the AD method s ntroduced. The model uses a sol surface energy balance equaton to determne sol surface boundary condtons for both heat and water flow. The nputs requred for the computer smulatons are weather data, sol thermal and hydraulc propertes, and mulch data. Numercal experments are performed to examne the effects of sol type, mulch wdth, and weather condtons on sol heat and water movement. For contnuous evaporaton and dranage, 10-day smulatons were performed for each combnaton of clay, loam, and sand sol and fractons of mulch cover of 0, 0.5, 0.8, and 1.0 of the row nterval wdth. For repettve evaporaton and nfltraton, 15-day smulatons were performed. The mulch cover greatly reduces evaporaton loss and the ampltude of daly sol temperature, water content, and pressure head varatons. Large spatal varatons n temperature and sol water content are predcted near the nterface of mulch and bare sol surface. The sol hydraulc propertes have mportant roles n controllng sol surface water content. The present model reasonably descrbes the sol thermal and hydrologc envronments and thus can be appled successfully n sol scence and groundwater hydrology and can be extended to related dscplnes. Dscplnes Agronomy and Crop Scences Fresh Water Studes Plant Bology Sol Scence Comments Ths artcle s publshed as Chung, Sang Ok, and Robert Horton. "Sol heat and water flow wth a partal surface mulch." Water Resources Research 23, no. 12 (1987): Do: /WR023012p Posted wth permsson. Rghts Accepted for publcaton n Water Resources Research. Copyrght 1987 Amercan Geophyscal Unon. Further reproducton or electronc dstrbuton s not permtted. Ths artcle s avalable at owa State Unversty Dgtal Repostory:

3 WATER RESOURCES RESEARCH, VOL. 23, NO. 12, PAGES , DECEMBER 1987 Sol Heat and Water Flow Wth a Partal Surface Mulch SANG-OK CHUNG 1 AND ROBERT HORTON Department of Agronomy, owa State Unversty, Ames A computer model usng the alternatng drecton mplct (AD) fnte dfference method to stud) two-dmensonal coupled sol heat and water flow wth a partal surface mulch cover s developed. A new, smplfed computatonal procedure, whch has only trdagonal matrx problems, for the AD method s ntroduced. The model uses a sol surface energy balance equaton to determne sol surface boundary condtons for both heat and water flow. The nputs requred for the computer smulatons are weather data, sol thermal and hydraulc propertes, and mulch data. Numercal experments are performed to examne the effects of sol type, mulch wdth, and weather condtons on sol heat and water movement. For contnuous evaporaton and dranage, 10-day smulatons were performed for each combnaton of clay, loam, and sand sol and fractons of mulch cover of 0, 0.5, 0.8, and 1.0 of the row nterval wdth. For repettve evaporaton and nfltraton, 15-day smulatons were performed. The mulch cover greatly reduces evaporaton loss and the ampltude of daly sol temperature, water content, and pressure head varatons. Large spatal varatons n temperature and sol water content are predcted near the nterface of mulch and bare sol surface. The sol hydraulc propertes have mportant roles n controllng sol surface water content. The present model reasonably descrbes the sol thermal and hydrologc envronments and thus can be appled successfully n sol scence and groundwater hydrology and can be extended to related dscplnes. NTRODUCTON A mulch nfluences the sol surface radaton balance, the sol water evaporaton rate, the sol temperature dstrbuton, and the mosture dstrbuton n the sol. The mulch can be effectvely used to reduce sol eroson n humd areas and to reduce water loss by evaporaton n ard areas. Several studes have been made on the effects of varous sol surface mulch- ngs on the temperature and (or) water dstrbuton n the sol. Mahrer [1979] studed one-dmensonal sol heat flow wth a transparent polyethylene mulch present. Mahrer and Katan [1981] dd research on two-dmensonal sol heat flow when a transparent polyethylene mulch constraned the evaporaton heat loss from a porton of the sol surface. Mahrer et al. [1984] studed one-dmensonal heat and water flow when a transparent polyethylene mulch covered the entre surface. Jury and Bellantuon [1976a, b] studed the sol heat and water envronment affected by rocks on the sol surface. Horton et al. [1984a, b] studed two-dmensonal sol heat transfer wth ncomplete sol surface plant cover. They used an explct fnte dfference method to solve the heat conducton equaton. The agreement between the predcted and observed temperatures n the sol profle was good. Most prevous studes were for one-dmensonal vertcal flow regons. n agrcultural practce, a partal sol surface mulch may be appled, and two-dmensonal models should be used for more realstc smulaton studes of heat and water transfer. n addton, because heat and water n the sol nteract wth each other, a coupled heat and water flow model should be used. Phlp and De Vres [1957] presented theory to descrbe coupled heat and water flow n sol. Van Bayel and Hllel [1975, 1976], Sophocleous [1979], and Mlly [1982] expanded and (or) used the theory to calculate heat and water flow n sol. An alternatng drecton mplct (AD) fnte dfference Now at Department of Agrcultural Engneerng, Kyungpook Natonal Unversty, Korea. Copyrght 1987 by the Amercan Geophyscal Unon. Paper number 6W /87/007W model of sol heat and water flow s developed to study the effects of a partal crop resdue mulch on the sol temperature and water dstrbutons n more detal than n prevous nvestgatons. The method s physcally based and general n that sol thermal and hydraulc propertes and standard meteorologcal data are the requred nputs. Ths model extends the model of Horton et al. [1984b] by ncludng water flow and by replacng the canopy shadng porton wth a partal surface mulch condton. Smulaton runs are made by usng varable sol, mulch, and weather condtons. Ten-day smulatons of sol heat and water flow are made for smultaneous evaporaton and dranage condtons, and 15-day smulatons are made for alternatng evaporaton, nfltraton, and dranage. The results of the selected smulaton runs are reported n graphcal and tabular forms for both the thermal and the hydraulc envron- ments. Flow Regon MATHEMATCAL MODEL The numercal model descrbes a system consstng of sol layer, mulch layer, and atmospherc layer. The crop resdue mulch strps are assumed to be parallel and equally spaced. A schematc descrpton of the flow regon s gven n Fgure 1, whch shows the cross secton perpendcular to the row drecton. Because of symmetry, one secton of the regon, secton ABCD, was consdered n the model study. A rectangular coordnate system s used wth orgn A at the upper left corner of the flow regon, x horzontally to the rght, and z vertcally downward. Flow Equatons The flow equatons governng the unsteady smultaneous heat and water flow were developed by Phlp and De Vres [1957] as follows: c3t C = V-(2VT)- LV.(Do, VO ) (1) c 0 c K - v. (tcvh) - (2) c t c z 2175

4 2176 CHUNG AND HORTON: SOL HEAT AND WATER FLOW where C s volumetrc sol heat capacty (J/m 3 øc), T s sol temperature (øc), t s tme (s), s thermal conductvty (W/m øc), L s volumetrc latent heat of vaporzaton (j/m3), 0 s volumetrc water content (m3/m3), Dov s sothermal vapor dffusvty (m2/s), K s hydraulc conductvty (m/s), h s pressure head (m), z s the vertcal dstance, postve downward (m), and V s gradent operator. n the present study, the effects of water vapor on heat and mosture transport are ncluded only at the sol surface. Subsurface vapor flow s not ncluded, thus the model s best appled to humd and subhumd regons where prolonged drought perods that manfest subsurface vapor transport are less frequent. Consequently, (1) was changed to: Equaton (2) can be modfed as follows: cqt C = V-(VT) (3) - T ' -[- h W = V' (KVh) --{ W (4) where h s pressure head (m) and K s hydraulc conductvty (m/s). n ths study the thermal lqud flow s assumed to be nsgnfcant, as demonstrated by Mlly [1984] for most sol water contents except very wet condtons, and (4) s reduced to AD Fnte Dfference Equatons Oh OK F -- = v- (KVh) - (5) at Oz Fnte dfference equatons can be derved by replacng the dfferentals n (3) and (5) by dfference expressons. For twodmensonal flow problems, the AD method has been used successfully [Selrn and Krkham, 1973]. n the AD method the fnte dfference equaton s set up usng one dmenson mplct whle leavng the other dmenson explct at a tme and then changng the drecton n the next tme step. The governng equatons (3) and (5) can be changed nto the AD fnte dfference equatons as follows: for the even traverse (z drecton). The superscrpts represent tme and steps, subscrpts space steps, the row ndex, and j the column ndex, and F s the specfc water capacty. Fgure 2 shows the fnte dfference dscretzaton of the flow regon. At each traverse, (6) and (7), and (8) and (9) have 2(M)(N) smultaneous equatons wth 2(M)(N) unknowns, T and h at each node, where M and N are the numbers of columns and rows n the flow regon, respectvely. n prncple, the equa- tons (6) and (7) for the odd traverse and (8) and (9) for the even traverse have to be solved smultaneously for each tme step. That means we have to solve a 2(M)(N) by 2(M)(N) matrx, whch requres a lot of computaton even though the matrx s banded wth a band wdth of sx n ths partcular case. To reduce the computaton requrements, a modfed procedure was followed by solvng each set of (M)(N) smultaneous equatons for each of (6) through (9) separately. Ths modfcaton wll not ntroduce large errors f a small tme step sze s used and wll greatly reduce necessary computaton nasmuch as, n each stage, (6) through (9) end up wth a trdagonal matrx that can be effcently solved by usng the so-called Thomas algorthm [Lapdus and Pnder, 1982]. Equatons (6) through (9) are nonlnear because the values of coeffcents are dependent on the values of the varables themselves. Therefore an teraton method can be used when solvng these equatons. However, f the tme step sze s small, the coeffcents n (6) through (9) can be approxmated by usng values from the prevous tme step such as 1/2 = "(10) 3/2 = n+ (11) Ths smplfes the computatons because the system of equatons becomes lnear. The computatonal procedure for determnng temperature and pressure head s as follows: (1) determne the values of coeffcents by usng values of varables at tme step n, (2) compute T "+ 1 usng T" (equaton 6), (3) compute h "+ 1 usng h" (equaton 7), (4) determne the values of coeffcents by usng values of varables at tme step n + 1, (5) compute T "+2 usng n+ 1/2 t..,j n t..,j n, J+ 1/2 n+ 1/2[T t',j+ n T,j n+ 1 ) --,j- 1/2 n+ 1/2(t x*,j ' n t, J - 1 n+ 1 ) C'j At = (Ax) 2 F,jn + 1/2,j,j l + 1/2,j n+ 1/2(T + 1,j - x *,j - (Az) 2 h n+l -h n K,j+ 1/2 n+ 1/2{h,",j+ n+ 1 h,j n+ 1) -- Kd- 1/2 n+ 1/211 U',j n hd- 1 n+ 1) At (Ax) 2 for the odd traverse (x drecton), and C jn + 3/2,J ' T.. n+2 T...n+ 1 At,d h n+2 n+l F " 3/2,j -- h,j,j At n+ 1/2t ' n n / n T,jn)_ l 1/2j -- T 1,j ) K + 1/2,j n+ 1/2(h + 1,jn h.j. )_ K - 1/2, n+ 1/2(h n - jn+ 1/2 K - n+ 1/2 (Az) 2,j+ 1/2 n+ 3/2(T/, j+ 1 n+ 1 T..,j n+ 1), J- 1/2 n+ 3/2(T..,j n+ T/j_, 1 n+ 1) (Ax) 2 1/2.5 n+ 3/2(T/+ 1. n T:.j n+ 2) -- 1/2.5 n+ 3/2(T...j n t/_ 1.5 n+. 2) (8) (Az) 2 K,j+ 1/2 n+ 3/2{h t",j+ 1 n+ -- h,j n+ 1) -- K,j- 1/2 n+ 3/2(h v-,j n+ -- h,j- 1 n+ 1) (Ax) 2 Kj+ 1/2,j n+ 3/2(h+ 1,j n+ 2 h,j,+ 2) K- 1/2,j n+ 3/2(h x-,j n h- 1, n+ 2) K+ 1, jn + 3/2 K_ 1,j n + 3/2 (Az) 2 2Az 2Az (6) (7) (9)

5 ,.. CHUNG AND HORTON' SOL HEAT AND WATER FLOW 2177 /A\v xx A t, B D Fg. 1. Schematc dagram of the flow regon. mulch T "+ (equaton 8), and (6) compute h "+ 2 usng h "+ (equaton 9). The nternodal coeffcents such as hydraulc conductvty and thermal dffusvty as shown n (6) through (9) were determned by usng the geometrc mean of the two neghborng nodes because t was shown that the geometrc mean gave more realstc results than dd the arthmetc mean [Haverkamp and Vaucln, 1979; Schnabel and Rche, 1984]. ntal and Boundary Condtons Specfc condtons, namely the ntal and boundary condtons, are needed to solve the flow equatons. The ntal condton ncludes varable values for each node n the flow sol regon at the begnnng of the smulaton. Boundary condtons can be ether one of a flux condton or a value- specfed condton, or the combnaton of the two. These boundary condtons are mathematcally termed Neumann condton, Drchlet condton, and Cauchy condton, respectvely. To effectvely handle the flux boundary condton s the most dffcult part n the fnte dfference method. Therefore most of the prevous studes used value-specfed boundary condtons, whch are smpler to handle than flux boundary condtons. A no-flow boundary condton s a specal case of the flux boundary condton and s much easer to handle than the nonzero flux boundary condton. n the present study, valuespecfed boundary condtons were used for the top and bottom boundary condtons for the heat flow, no-flow boundary condtons for rght and left boundares for both heat and water flow, and nonzero flux boundary condtons for top and bottom boundares for water flow. The sol surface boundary condtons for both heat and water flow are not explctly known (nput), but are mplctly determned by energy parttonng (dscussed n Energy Balance Equaton secton). There s a specal requrement to keep a hgh accuracy n the AD method for the ntermedate tme step boundary condtons for tme dependent boundary condtons as descrbed by Lapdus and Pnder [1982, pp ]. However, that requrement can be satsfed only for explct value-specfed boundary condtons. n ths study, snce the boundary condton, ether value or flux condton, s mplctly determned by energy balance equaton, the specal requrement cannot be satsfed. At the ntermedate tme step, the boundary condton at the prevous tme step was assumed. When employng value-specfed boundary condtons, the value at each specfed tme s placed drectly nto the fnte dfference equaton. To handle the flux boundary condtons n the fnte dfference method, magnary nodes are ntroduced outsde the flow regme [Lapdus and Pnder, 1982' Gldng, 1983]. For example, the flux at node 1 can be expressed n fnte dfference form as flux = -- A(H 2 -- Ho)/2Az (12) where A s thermal or hydraulc conductvty and H s total head or temperature. Subscrpt 0 represents an magnary node outsde node 1. For a no-flux boundary, (12) equals zero, hence H 2 = H o. Therefore an expresson for the magnary pont, H o, can be smply replaced by H 2 n (6) to (9). For a nonzero flux boundary, solve (12) for the magnary pont, H o, then replace H o, whch s expressed n terms of H 2 and the flux, n (6) to (9). The flux at node N can be handled smlarly. The sol surface temperature and evaporaton rate were determned by usng an energy balance equaton for the sol surface durng dry weather. nfltraton rate and sol surface temperature durng the rany weather were determned from Darcy's equaton and ar temperature, respectvely. n the energy balance approach, there s no need to worry about the potental and actual evaporaton rates because the method drectly calculates actual evaporaton. However, the potental nfltraton rate s governed by ranfall amount, whereas the actual rate s governed by both ranfall amount and sol nfltrablty. Those two should be compared, and the smaller governs the actual nfltraton rate. A unt gradent of total head was used for the water flow bottom boundary condton. Energy Balance Equaton Sol surface temperature and evaporaton rate were determned mplctly from the parttonng of the surface energy. A procedure descrbed by Van Bavel and Hllel [1975, 1976] and -l' l - N=21 z Fg. 2. (,j) j-1 j j+l L = 0.35m 1 q M=8 Fnte dfference dscretzaton of the flow regon. J L 2 = 1.0m

6 2178 CHUNG AND HORTON:SOL HEAT AND WATER FLOW Horton b] was used. The energy balance at the sol surface for bare sol s descrbed by R. -- H s-- LE -- G = 0 (13) where R, s net radaton (postve downward), H s s sensble ar heat flux (postve upward), LE s latent heat flux (postve upward), and G s sol heat flux (postve downward). The value of R, s found as R,, = (1-- al)r + R 1 -- eo'(t s ) ½ (14) where R (W/m 2) s the measured global radaton, Rl(W/m 2) s the long-wave sky rradance, T s the surface temperature (øc), al s the sol surface albedo, e s the emssvty, and cr(w/m 2 øk ) s the Stefan-Boltzmann constant. R 1 s calculated as was done by Van Bayel and Hllel [1976] from the followng form of Brunt's formula: R = cr(t, ) [ (1370 H,) a2] (15) where T, s ar temperature (øc) and H, s the ar humdty (kg/m3). The latent and sensble heat fluxes at the surface were calcu- lated by usng the followng equatons: E = (m 0 -- m.)/(l()00 %) (16) L = (109) (106)T (17) LE = L E (18) H s - p c, (T - T )/% (19) where E s the evaporatve flux (m/s), L s volumetrc latent heat of vaporzaton (J/m 3) gven by Forsythe [1964], H 0 s the absolute humdty of ar at the sol surface (kg/m3), H. s the absolute humdty of ar above sol surface (kg/m3), r. s the aerodynamc boundary layer resstance between the sol surface and the ar above t (s/m), p s ar densty (kg/m3), c,. s specfc heat of ar at constant pressure (J/kg øc). The absolute humdty, H0, and aerodynamc resstance were calculated by usng the followng equatons [Van Bayel and Hllel, 1976]: H o = Ho* exp [h /46.97(T )] (20) r, = [n (2.0/Zo)J2/O. 16W (21) where H0* s the saturaton humdty at the sol surface temperature (kg/m3), h s the pressure head at the surface (m), Z 0 s roughness length (m), and W s the wnd speed (m/s). n (20), H 0 depends not only upon the surface temperature but also on the surface water content, and h cannot be greater than zero n (20). The absolute humdty of ar, H,, and the saturaton humdty at the sol surface temperature, H0*, were calculated by the followng equatons: H, = exp [17.27 Td(T )]/(T, ) (22) H0* = exp [17.27 T /(T )]/(T ) (23) where T a s the dewpont temperature (øc). The sol heat flux at the sol surface was determned by G=, T --T2 Az Az + PsCps(T - T ø) 2A (24) where, s thermal conductvty (W/m øc), T s unknown temperature on the sol surface (øc), T ø s T at prevous tme step (øc), T 2 s temperature at node Az below the sol surface at prevous tme step (øc), Ps s sol densty (kg/m3), cs, s s specfc heat of sol at constant pressure (J kg/øc), Az s step sze n z drecton (m), and At s tme step (s). Equaton (24) approxmates the sol heat flux densty by summng a term that estmates sol heat flux at a depth of Az/2 and a term that estmates the change n heat stored n the sol above Az/2. The second term on the rght-hand sde s a correctve term to compensate the error n the frst term for the relatvely large step sze of Az. For a mulched surface, addtonal consderaton should be gven. The energy balance equaton should be appled at both the mulch surface and at the mulch-sol nterface as descrbed by Van Bayel and Hllel [1975]. n ths analyss we assume nontransparent mulch cover such that radaton does not penetrate below the surface. For the top of the mulch surface, the energy balance equaton s R.- H s-- M s = 0 (25) where R, and H s are the same as (13) and M s s the mulch heat flux (postve downward). The H s can be determned as a s = pa%a(tm- Ta)/r (26) M s = ;%(T m -- T )/THK (27) where T m s temperature on the mulch surface (øc),,,. s thermal conductvty of mulch layer (W/møC), THK s the thckness of mulch layer (m), and the others are the same as prevously defned. For the mulch-sol nterface, the energy balance equaton s M s -- LE -- O = 0 (28) The latent heat flux on the sol surface was calculated by usng (17) and (18) wth E = (H o -- H.)/[lOOO(r + rm) ] (29) where all are the same as n (16) wth an addtonal term rm, the dffuson resstance (s/m) of the mulch. The r m s determned by [Hllel et al., 1975]: r m -- THK/Datmfr (30) where THK s the thckness of the mulch layer, Dat m s the vapor dffusvty n ar (m2/s), f s the mulch porosty, and r s tortuosty factor. Ths analyss does not consder convectve transport of gas wthn the mulch layer, whch, f present, wll act to decrease r m. To determne the sol surface temperature and evaporaton rate, the energy balance equaton was solved. For bare sol, (14), (18), (19), and (24) were substtuted nto (13). All these equatons are unknown functons of T. Therefore an teratve root-fndng method s used to solve Eq. (13) for T (the surface temperature). The bsect method was used for root fndng. The T s value from the prevous tme step was used as an ntal guess, and the teraton was contnued untl the dfference of T n successve teraton was less than a predetermned tolerance. When T s was determned, the evaporaton rate was also determned by (16). For mulched sol, (14), (26), and (27) are substtuted nto (25). Then, the same procedure as n the bare sol was followed to determne the mulch surface temperature, Tm, and mulch heat flux, M s. Wth ths M s as an nput, (17),(24),(27), and (29) are substtuted nto (28) to determne the mulch-sol nterface temperature, T. The same teratve procedure as n the bare sol was followed to determne T. As soon as T s determned, the evaporaton rate s also determned by (29).

7 CHUNG AND HORTON' SOL HEAT AND WATER FLOW 2179 nputs Requred for Computer Smulaton Specfc weather, mulch, and sol parameters are requred as nputs to solve sol heat and water flow problems. The weather nputs are daly global radaton, maxmum and mnmum ar temperature, maxmum and mnmum dewpont temperature, and average daly wndspeed. The weather nputs are those at the heght of 2 m above the ground. The mulch nputs are wdth, thckness, thermal conductvty, mosture dffuson resstance, porosty, and tortuosty factor. Sol parameters requred as nputs are ntal temperature and water content dstrbutons, lower boundary temperature and water content as a functon of tme, sol surface emssvty as a functon of water content, sol surface albedo as a functon of water content, sol thermal dffusvty as a functon of water content, sol water hydraulc conductvty as a functon of water content, specfc water capacty as a functon of water content, and the sol-water characterstc curve. A few general nputs are also requred to run the computer program. The nputs are as follows: L, L2, Az, Ax, solar noon, daylength, tme length of smulaton, At (ncremental tme step), and Z o (the surface roughness length). The weather nputs are used n conjuncton wth emprcal expressons to descrbe the weather condtons as a functon of tme. Global radaton as a functon of tme for sol n drect sunlght s descrbed as R 0 = (n/2) DR/DL sn [(t -- SN + DL/2) r /DL] (31) where R0 s n W/m 2, DR s daly global radaton (j/m2), t s tme of a day (s), SN s solar noon (s), and DL s daylength (s). Equaton (31) dstrbutes the daly radaton durng the daytme by usng a sne functon. The ar temperature and dewpont temperature were also determned by sne functons as follows: T a = Ta + A sn (2nt/ n) (32) T = T + A d sn (2nt/ n) (33) where the bar represents the average and A represents ampltude, and t s tme of a day begnnng from mdnght (s). The n was ncluded to allow the hghest ar temperature to be at noon. Sol surface emssvty, e, follows that used by Van Bayel and Hllel [1976], (34) Sol surface albedo, al, follows that used by Van Bavel and Hllel [ 1976], al = < 0 < 0.25 al = < 0 al = < 0.10 (35) Thermal conductvty s descrbed by a smple emprcal equaton' 2(0) = b + b20 + b30 ø'5 (36) where 2 s thermal conductvty (W/m øc), 0 s volumetrc water content (m3/m3), and b, b2, and b 3 are the regresson parameters. The volumetrc sol heat capacty s determned by followng De Vres [1963] as C= 1.92 x 106x s+2.51 x 106x o+4.18 x 1060 (37) Where C s n J/m 3 øc, x s and x o are volume fractons of sold and organc matter n the sol, and 0 s the volumetrc water content. Sol water characterstcs, hydraulc conductvty, and specfc water capacty are descrbed by emprcal equatons presented by Van Genuchten [1980] as follows: 0 = 0,. + (O s -- 0,.) 1 + ah) 1 - (l/n) ' (1 -- (ah)"-l[1 + (ah)"] -'ø/"} 2 (38) K(h) = K s [1 + (ah)"] "- a /2, (39) (0, n) = (n- 1)(0-0,) (40) where O s and 0 r are saturated and resdual water content, K s s saturated hydraulc conductvty at the reference temperature, h s absolute value of pressure head, and a and n are nonlnear regresson parameters descrbng the shape of the sol water characterstc curve. The hydraulc conductvty should be corrected for temperature as: K(h, T)= K(h)Kt(T) (41) where K,(T) =/ (To)//a (T), the temperature correcton factor,/ s the vscosty, and T O s the reference temperature. Here the densty effect s consdered neglgble compared to the vscosty effect. TEST OF MODEL The performance of a numercal model should be evaluated to examne ts valdty because any numercal scheme may ntroduce nstablty, truncaton, and round-off errors. A model s vald only f the approxmate soluton s satsfactorly accurate or close to the exact soluton f one exsts. The accu- racy of a model can be more specfcally defned n terms of ts convergence and stablty. Convergence s satsfed when the approxmaton approaches the exact soluton as step szes of the spatal and temporal dscretzaton approach zero. A model s sad to be stable f the amplfcaton of the error s restrcted or has a fnte lmt as computaton marches forward n tme. The valdty of a model can be tested by comparng the numercal soluton wth ether an analytcal soluton, f t s avalable, or observed data. Snce there s nether an analytcal soluton nor measured data for the two-dmensonal smultaneous heat and water flow, the entre model developed here cannot be tested drectly aganst an analytcal soluton or measured data. Therefore, n ths paper, only the heat flow part of the AD method s tested through comparson wth an analytcal soluton. The heat conducton equaton for a hot steel rod of semnfnte length wth a rectangular cross secton beng exposed to a coolng ar stream was solved numercally usng the AD method and then compared wth the analytcal soluton gven by lncropera and DeWtt [1981, pp ]. Fgure 3 shows the cross secton area of the steel rod. Snce the conducton heat flow s symmetrcal n both drectons, only one quarter of the cross secton s consdered n the analyss. Fgure 3 also shows the x and z coordnate system and the boundary condtons of noflux condtons along the symmetrcal lne and flux condtons along the outsde boundares. The dmenson and thermal propertes of steel rod used are 0.5 by 1.0 m, thermal conductvty of 20 W/m øk, densty of 3000 kg/m s, and specfc heat

8 2180 CHUNG AND HORTON: SOL HEAT AND WATER FLOW 1.0m L O. 5m 3 T TABLE 1. Hydraulc and Thermal Propertes of the Sols Parameter* Clay Loam Sand K, (m/s) 0.2 x x x 10-5 O s (m3/m 3) O r (m3/m 3) a (m-t) n (38) n n (38) bt n (36) b 2 n (36) b 3 n (36) *K, s saturaton hydraulc conductvty, 0 s s saturaton water content, 0 r s resdual water content, a and n are parameters n Van Genuchten's retenton equaton, and b t, b2, and b 3 are parameters n thermal conductvty equaton. xx T =0 : z _k T=h z (T-Tar) _k _ _=T x h ( (T-Ta r) Fg. 3. Cross-secton, flow regon, and boundary condtons for the heat flow problem n a steel rod. of 1000 J/kg øk. The convecton heat transfer coeffcent of the ar stream s 10 W/m 2 øk. The ntal temperature of the steel rod s 300øC, and the ar stream temperature was mantaned at 20øC. ^ spatal step sze of 0.01 m and the tme step sze of 5 s were used. A smulaton run was made for 4000 s durng whch the temperature of the rod decreased near the ar stream temperature. Fgure 4 shows the temperature dstrbuton n the steel rod at the selected tmes as determned analytcally and numercally. The AD method s shown to compare favorably wth the analytcal method. Even at later tmes as the temperature n the steel rod approaches the ar stream temperature, the agreement between the two solutons s good. O.O O '// (a) at t=520 s (b) at t=1040 s Fg. 4. Comparson of solutons of the heat flow n a steel rod between AD soluton (sold lne) and analytcal soluton (dashed lne). NUMERCAL EXPERMENTS Smulaton runs were made usng varable sol, mulch, and weather condtons. Three hypothetcal sols, representng a sand, a loam, and a clay were selected for the smulatons. Varable wdth of crop resdue mulch cover and varable weather condtons were consdered. Sols were assumed to be homogeneous and sotropc. The sol water retenton and hydraulc conductvty relatons of these sols were obtaned from Hllel and Van Bayel [1976]. Retenton data were pcked up from the curves gven n Hllel and Van Bavel [1976] and were used to determne regresson parameters n Van Genuchten's retenton equaton (equaton (38)) by graphcally curve fttng as explaned n Van Genuchten [1980]. Prevously reported data were used to descrbe the thermal propertes of these sols. To determne the emprcal parameters n the thermal conductvty equaton (equaton (36)), data n Table 7.6 of De Vres [1963] and n the works by Werenga et al. [1969] and Horton and Werenga [1984] were used for clay, loam, and sand, respectvely. Table 1 shows the thermal and hydraulc parameters for the three sols. A mulch may vary n thckness, wdth, and materal. n the present study, a m-thck crop resdue (corn) mulch wth varable wdth was used. A row nterval of 0.70 m was used n ths study. Mulch wdths of 0, 0.35, 0.55, and 0.70 m were used n the smulatons. The dffuson resstance of the mulch was determned by (29), and the value of 1200 s/m was obtaned. Ths s about an order of magntude larger than the aerodynamc resstance between the sol surface and the ar, %. Varous weather condtons can be used wth the numercal model. n the present study, frst, 10-day smulatons of a dry weather condton (no ran), allowng smultaneous dranage and evaporaton, were performed. Second, 15-day smulatons of repeatng dry and rany weather, allowng smultaneous dranage and ether evaporaton or nfltraton were performed. The ranfall occurred at the begnnng of the thrd day and lasted 6 hours: 2 hours at a low steady ntensty of 10 mm/hr, 2 hours at a hgh steady ntensty of 20 mm/hr, and 2 hours at a low steady ntensty of 10 mm/hr, for a total ranfall of 80 mm. An ntal condton of 20øC was used for sol temperature, and -1.1 m of sol surface pressure head wth a sol-profle equlbrum condton was used for water. A bottom boundary condton of 20øC was mantaned for heat flow, and a gravty flux condton was mantaned for the water flow. Average daly weather nput data (except for ranfall) for Des Mones, owa, n the month of June were used as shown by Van Bavel and Hllel [1976]. The flow regon has wdth of 0.35 m, one half of the row

9 CHUNG AND HORTON: SOL HEAT AND WATER FLOW 2181 TABLE 2. nput Parameter Values Used n the Smulatons Parameter Defnton Value DAYL Daylength s DELX Spatal step sze n x 0.05 m coordnate DELZ Spatal step sze n z 0.05 m coordnate RAMDAM Thermal conductvty of mulch W/m øc RGD Daly global radaton x 106 J/m 2 RLENG Wdth of mulch cover Varable RM Mulch mosture dffuson 1200 s/m resstance SNOON Solar noon s TAVE Average daly ar temperature 21øC TAMP Ampltude of daly ar temperature 5øC TAW Mulch tortuosty factor 0.67 TDAMP Ampltude of daly dewpont 2øC temperature TDEW Average of daly dewpont 16.4øC temperature THKM Mulch thckness m TMEST Tme step sze Varable WS Wnd speed 1.69 m/s XLENG Length of x coordnate 0.35 m ZLENG Length of z coordnate 1.0 m ZO Sol surface roughness length 0.01 m nterval, and depth of 1.0 m. A spatal step sze of 0.05 m was used for both drectons. Tme step szes of 300 and 600 s were used for sand and the others, respectvely, for the contnuous dry weather. Durng and for some perod after a ranfall the tme step sze was reduced to 1/10 to 1/200 that of dry weather. Table 2 shows the nput parameter values used n the smulaton. The numercal smulaton provdes several descrptons for TABLE 3. Daly Totals of Net Radaton (Rn), Sensble Heat (S 0, Latent Heat (LE), and Sol Heat (G) and the Maxmum and Mnmum Temperature on the Bare Sol Surface (Node 8) of the Sand Sol Wth a Half-Wdth Mulch Cover R., S h, LE, G, T... T.m., Day MJ/m 2 MJ/m 2 MJ/m: MJ/m 2 deg C deg C both the thermal and the hydraulc envronments. ncluded for the thermal envronment are sol temperature dstrbuton, nstantaneous and cumulatve net radaton, sensble heat flux, latent heat flux, and sol heat flux at the sol surface. ncluded for the hydraulc envronment are pressure head and water content dstrbuton, nstantaneous and cumulatve nfltraton, and evaporaton, dranage, and storage of water n the flow regon. RESULTS AND DSCUSSON A two-dmensonal sol heat and wate flow model was run for 24 partcular sets of nput parameters (three sols, two weather regmes, and four surface mulch wdths). nput weather parameters (except ranfall) were allowed to recycle n the same manner for each day of the smulaton perod. The results of the computer-smulated thermal envronment and hy- DE 8 (a) (b) DE - ' ' ' ' ' ' ' ' TME [DRY} t0 TME (DRY) NODE lb NODE B (c) (d) "')o _ NODE.g :2:. TME (DRY) TME (DRY) Fg. 5. Predcted cumulatve heat flux of net radaton (a), sensble heat (b), latent heat (c), and sol heat (d) for the sand sol wth a half-wdth mulch cover for a 10-day smulaton.

10 2182 CHUNG AND HORTON' SOL HEAT AND WATER FLOW TABLE 4. Daly Totals of Net Radaton (R,), Sensble Heat (Sh), Latent Heat (LE), and Sol Heat (G) on Day 1 and 10 for Full Mulch and Zero Mulch (Bare) Condtons Wth Smultaneous Evaporaton and Dranage for a 10-day Smulaton Day 1 Day 10 R n, Sh, LE, G, R n, Sn, LE, G, Sol Cover MJ/m 2 MJ/m 2 MJ/m 2 MJ/m MJ/m 2 MJ/m: MJ/m MJ/m: Clay Mulch Bare Loam Mulch Bare Sand Mulch Bare 13, draulc envronment wth partcular focus on two representatve smulatons are gven n ths secton. The focus s on the sand sol wth a half-wdth mulch cover wthout ranfall. Results from ths smulaton are dscussed to dsplay the ablty of the model and to show the major nfluence of a partal mulch cover on heat and water flow. The results from the other smulatons are presented n a manner to show dfferences and smlartes relatve to the smulaton that receved the major focus. Thermal Envronment The results of selected smulaton runs are presented n graphcal form for heat flow. The 10-day smulatons of smultaneous evaporaton and dranage for the sand sol are dscussed. Fgure 5 shows the predcted cumulatve heat fluxes at the sol or mulch surfaces for the sand sol wth one half of the row nterval covered wth mulch durng a 10-day smulaton. Nodes 1 and 8 represent the mddle of the mulch strp and the mddle of the bare sol strp, respectvely. At node 1 wth mulch cover, net radaton and sensble heat flux are at the mulch surface, and latent heat and sol heat flux are at the bare sol-mulch nterface. The cumulatve net radaton on bare sol s about twce as large as that on the mulch, partly because of the larger albedo on the mulch. At node 1 on or under the mulch, all the cumulatve heat flux curves mantan constant trend slopes for the entre perod, whereas at node 8 on a bare surface, these curves change from day 5 (Fgure 5). From day 5, net radaton decreases, sensble heat and sol heat flux ncrease, and latent heat flux becomes zero. Ths ndcates that from day 5, lttle sol surface water s avalable for evaporaton. Snce the daly weather nput values are the same for the smulaton perod, the change n heat flux from day 5 s caused by changng sol thermal propertes and the parttonng of the energy formerly used for evaporatng water. The rapd ncrease n cumulatve sol heat results n very hgh sol surface temperatures on the bare surface. Fgure 6 shows temperature varaton at the sol surface and õ SOL SURF!CE NODE SOL SURFlCE NODE 8 (a) o (b) o 4/ '" 6!,, 110 TME (DRY) o m c! TME [BRT) m m CM DEPTH NODE õ 5 CH DEPTH NODE 8 (c) 4/5, 1/2 o (d) L œo o 112 4/5 [ 1[0 TME (DRY) o TME tory) Fg. 6. Predcted sol temperature at the sol surface node 1 (a) and node 8 (b), and at the 5-cm-depth node 1 (c) and node 8 (d), for the sand sol wth varable mulch cover wdth.

11 ß CHUNG AND HORTON' SOL HEAT AND WATER FLOW õ 25 DAY 1, 2:00 p.m. (a) DAY 5, 2:00 p.m. temperature (øc) water content (m3/m 3) pressure head (m) (b) Fg. 7. Contour plots of temperature, water content, and pressure head n the 0.35 x 1.0 m flow regon at 2.00 P.M. for day 1 (a) and day 5 (b) for the sand sol wth a half-wdth mulch cover ß O the 5-cm depth wth varable mulch cover wdth. At the sol surface node 1, the trend slopes of temperature varaton change from day 5 and from day 9 for a half- and a 4/5-wdth mulch, respectvely. These correspond to a sudden ncrease n temperature at node 8 n Fg. 6b. Ths confrms the exstence of lateral heat flow fro m the bare sol to the mulched sol for partly mulch covered sols. Sol temperature the 5-cm depth responded n a manner smlar to the sol surface except wth smaller temperature ampltudes. Fgures 6b and 6d show the effect of mulch wdth on the sol temperature varaton. Abrupt ncreases sol temperature at node 8 occur at day 4, day 5, and day 9 for zero and a half- and a 4/5 wdth mulch cover, respectvely. These abrupt changes are coupled to the changng sol surface water contents. Temperature and mosture content changes at the plant seed poston (5-cm-depth, node 8) due to the mulch cover are mportant. The daly maxmum and mnmum temperatures the 5-cm depth (node 8) wth a half-wdth mulch cover show no dfference from those wth no mulch untl day 3, and both show 5øC lower than those wth no mulch from day 4 for the sand sol durng a 10-day smulaton. Wth a 4/5-wdth mulch cover, daly maxmum and mnmum temperature of the 5-cm depth (node 8) are øc lower and 0.5øC hgher, respectvely, than those wth a half-wdth mulch cover untl day 3, and are 8øC lower and 4øC lower, respectvely, than those wth a halfwdth mulch cover from day 4. For the sand sol, the temperature at the poston where the plant seeds are located s very much affected by the wdth of mulch cover. As the mulch cover wdth ncreases, the temperature at ths poston decreases. Ths mght ncrease the tme requred for germnaton of sprng plants. Fgure 7 shows contour plots of temperature, water content, and pressure head at 2:00 P.M. for the sand sol wth a halfwdth mulch cover. On day 1, temperature on the sol surface SOL SURFRCE NODE SaL SURFRCE NODE 8 (a) (b) o 4/5 ' TME S DF¾. TME (DAY),, 5 CH DEPTH NODE oo S CH OEPTH NODE 8 õ (c) g (d) :Z ' -"--'----- O / r - TME (DRY) Fg. 8. Predcted water content at the sol surface node 1 (a) and node 8 (b), and at the 5-cm-depth node 1 (c)and node 8 (d), for the sand sol for varable mulch cover wdth. TME D T)

12 2184 CHUNG AND HORTON' SOL HEAT AND WATER FLOW (a) (b)? ME {OR¾} SRNO o 5 1o TME {DRY) Fg. 9. Predcted water storage n the 0.35 x 1.0 m flow regon n unt.thckness(a) and cumulatve dranage across the bottom boundary (b) for the sand sol wth varable mulch wdth. changes rapdly near the bare sol-mulch nterface on the sol surface as shown n Fgure 5a. At day 5, temperature contour lnes near the rght, upper corner are very dense, ndcatng rapd temperature change near the corner. Bare sol had hgh temperatures. Beyond day 5 the temperature dstrbuton at 2:00 P.M. on each remanng day does not change much from Fgure 7b. Table 3 shows the daly totals of the components of the surface energy balance and the maxmum and mnmum temperature on the bare sol surface (node 8) of the sand sol wth a half-wdth mulch cover. Smulatons were made for repettve evaporaton and nfltraton on the sol surface. Ranfall started at the begnnng of day 3 and lasted for 6 hours wth a total ranfall amount of 0.08 m. Durng the ranfall event, sol surface temperature was set equal to the ar temperature. Though the outputs are not shown, the results of smulatons wth ranfall show that the ranfall reduces the prevously very hgh sol surface temperature by supplyng water for evaporaton on the sol surface and that the ranfall delays the tme perod requred for the surface to be very dry and to have hgh temperature. Table 4 summarzes the daly total energy balances on day 1 and day 10 for each sol wth or wthout mulch cover for the 10-day smulatons of smultaneous evaporaton and dranage. The components of the energy balance equaton for mulched sol have nearly the same values regardless of sol type durng the 10-day smulatons. There are some dfferences n the values of the components for the bare sol surfaces. Energy parttonng was smlar on days 1 and 10 for the clay bare surface. The sand and loam sol surfaces were dred by day 10, and the energy prevously used for latent heat was shfted to sensble and sol heat. Hydraulc Envronment The results of the sol heat and water movement smulaton runs relatng to the hydraulc envronment are reported n ths secton. The results nclude sol water storage, dranage, evaporaton, sol water content, and pressure head. The results of the 10-day smulatons for the sand sol wth a half-wdth mulch cover are reported. Fgure 8 shows the water content at the sol surface and the 5-cm depth for the sand sol wth varable mulch wdth. The resdual water content s reached and mantaned at node 8 on the surface after days 3, 4, and 8 for a zero-, a half-, and a 4/5 wdth mulch, respectvely. At the 5-cm depth node 8, water content frst decreases to 0.15, then ncreases, as shown n Fgure 8d. Ths s caused by reduced upward water loss due to the very small hydraulc conductvty near the sol surface whle the sol receves mosture from below, resultng n postve net water flux at ths poston. Fgure 9 shows the sol water storage and the cumulatve dranage for the sand sol wth varable mulch wdth. Fgure 9a shows that the fracton of mulch cover affects the sol water storage. Sol water storage s affected by mulches manly through the effect of mulch cover on sol water evaporaton. Fgure 9b shows that the rate of dranage across the bottom boundary decreases exponentally n all of the mulch condtons. Contour plots of sol water content and pressure head for the sand sol wth a half-wdth mulch are shown n Fgure 7. Near the bare sol surface the water contents are close to the resdual water content, and the sucton heads are very large from day 5. Table 5 shows the daly totals of evaporaton, dranage, and the maxmum and mnmum water content and pressure head on the bare sol surface (node 8) of the sand sol wth a halfwdth mulch cover. Ranfall s an mportant contrbutor to sol water movement. n ths study, all the 0.08 m of ranfall for the 6 hours on day 3 s nfltrated nto the sol. Though the outputs are not shown, the ranfall changes water content and pressure head near sol surface abruptly. t rases the evaporaton rate and the volume of water stored n the flow regon. t also ncreases the dranage rate when the nfltrated water reaches the bottom boundary. Table 6 summarzes the daly total water balance for dffer- TABLE 5. Daly Totals of Evaporaton (E), Dranage (D), and the Maxmum and Mnmum Water Content and Pressure Head on the Bare Sol Surface (Node 8) of the Sand Sol Wth a Half-Wdth Mulch Cover mn, h... hmn, Day E, mm D, mm m3/m 3 m3/m 3 m m

13 CHUNG AND HORTON: SOL HEAT AND WATER FLOW 2185 TABLE 6. Daly Total Water Balance n the 0.35 x 1.0 m Flow Regon Wth Unt Thckness on Day 1 and 10 for Varous Surface Mulch Wdths Wth Smultaneous Evaporaton and Dranage for a 10-Day Smulaton Day 1 Day 10 Evaporaton, Dran, Storage,* Evaporaton, Dran, Storage,* Sol Mulch 10-3 m m m m m m 3 Clay / / Loam / / Sand / / *At the end of each day. ent sol surfaces n the 0.35 x 1.0 m flow regon n unt thckness on day 1 and day 10 for the 10-day smulatons. As the fracton of the mulch wdth becomes larger, the daly evaporaton becomes smaller as long as enough mosture s suppled to the sol surface. On the other hand, as the fracton of the mulch wdth becomes larger, the daly dranage becomes greater also. However, the effect of mulch cover on evaporaton seems more mportant than dranage for causng dfferences n water storage. The effect of mulch cover on water storage at the end of the 10-day smulatons was greater for the clay and loam sols than for the sand sol. CONCLUSONS An AD method was developed and used to make predctons of sol surface energy partonng and of sol heat and water movement n a two-dmensonal flow regon wth partal sol surface mulch cover. The conclusons based on the present smulaton study are as follows: 1. The net radaton heat flux s much smaller and the sensble heat flux s much greater on the mulch than on the bare sol surface when the sol surface s relatvely wet. 2. The net radaton energy s parttoned manly to latent heat as long as the sol surface s relatvely wet. t s used manly for sensble and sol heat when the sol surface s dry. 3. The sol surface temperatures are very hgh when the sol surface approaches ts resdual water content because ncomng net radaton s used to heat up the sol surface nstead of to evaporate sol surface mosture. 4. The ampltudes of daly temperature, water content, and pressure head varaton under the mulch are much smaller than those on the bare sol surface. 5. The ampltudes of daly temperature, water content, and pressure head varaton decrease rapdly as sol depths n- crease. 6. The sol heat and water flow s nearly one-dmensonal below a sol depth of 40 cm. 7. The lateral heat and water flows near the sol surface wth a partal mulch cover are sgnfcant. 8. The mulch cover suppresses the sol water evaporaton to a large extent. 9. The mulch cover has only small mpact on sol water dranage. Therefore, the mulch effect on the storage n the flow regon s mostly governed by evaporaton. 10. The partal mulch cover does not have a large effect on the water content at the 5-cm depth (node 8), where the plant seeds are located, durng the 10-day smulaton perods. 11. The mulch cover wdth effects on the 5-cm-depth, bare (node 8) sol temperature are small n the clay and loam sol and are large n the sand sol durng a 1 O-day smulaton. 12. The changes n the sol thermal and hydraulc envronments are most rapd n the sand sol, followed by the loam sol and the clay sol. The present AD model has a wde applcaton n sol scence and groundwater hydrology and can be extended n many dfferent ways. The excluson of a vapor phase transport from the heat and water flow model may cause some dscrepances from the actual sol systems, especally for very dry sol condtons. Ths must be studed n the future. NOTATON A ampltude of daly temperature varaton (øc). a parameter n water retenton equaton (m- ). al sol surface albedo. C volumetrc sol heat capacty (J/m 3 øc). %a specfc heat of ar at constant pressure (J/kg øc). c, specfc heat of sol at constant pressure (J/kg øc). Dov sothermal vapor dffusvty (me/s). DL day length (s). DR daly global radaton (J/me). E evaporaton rate (m/s). F do/dh, specfc water capacty (m- ). f porosty of mulch (m3/m3). G sol heat flux (W/me). g gravtatonal acceleraton (m/se). H relatve humdty. H a ar humdty (kg/m3). H 0 ar humdty at sol surface (kg/m3). Ho* saturaton humdty at the sol surface temperature (kg/m3). H s sensble ar heat flux (W/me). h pressure (sucton) head (m). K hydraulc conductvty (m/s). K s saturated hydraulc conductvty (m/s). L volumetrc latent heat of vaporzaton (j/m3). LE latent heat flux (W/me). M s mulch heat flux (W/me). n parameter n water retenton equaton.

14 2186 CHUNG AND HORTON: SOL HEAT AND WATER FLOW Rg global radaton (W/m2). R n net radaton (W/m2). R long-wave sky rradance (W/m2). r, aerodynamc boundary layer resstance (s/m). r m the dffuson resstance of mulch layer (s/m). $N solar noon (S). T temperature (øc). t tme (s). T, ar temperature (øc). T a dewpont temperature (øc). T m mulch surface temperature (øc). T sol surface temperature (øc). THK thckness of mulch (m). W wnd speed (m/s). x horzontal coordnate, postve to the rght (m). x s volume fracton of solds n the sol. x 0 volume fracton of organc matter n the sol. z vertcal coordnate, postve downward (m). Z 0 sol surface roughness length (m). o thermal dffusvty (m2/s). Ax x drecton space step sze (m). Az z drecton space step (m). At tme step sze (s). e emssvty. 0 volumetrc water content (m3/m3). 0 volumetrc lqud water content (m3/m3). 0 r resdual water content (m3/m3). 0 s saturaton water content (m3/m3). ;t thermal conductvty of sol (W/m øc). ;t m thermal conductvty of mulch layer (W/m øc). p, densty of ar (kg/m3). Ps densty of sol (kg/m3). a Stefan-Boltzmann constant (W/m 2 øk' ). tt vscosty of lqud water (pose). co angular frequency (rad/s). V gradent operator. Acknowledgments. Journal paper J of the owa Agrculture and Home Economcs Experment Staton, Ames, owa, Projects 2556 and REFERENCES De Vres, D. A., Smultaneous transfer of heat and mosture n porous meda, œos Trans. AGU, 39, , De Vres, D. A. Thermal propertes of sols, n Physcs of Plant Envronment, edted by W. R. Van Wjk, North Holland, Amsterdam, Forsythe, W. E., Smthsonan physcal tables, Smthson. nst. Publ., 4169, 827, Gldng, B. H., The sol-mosture zone n a physcally-based hydrologc model, Adv. Water Resour., 6, 36-43, Haverkamp, R., and M. 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Horton, Department of Agronomy, owa State Unversty, Ames, A (Receved February 9, 1987; revsed September 2, 1987; accepted September 8, 1987.)