Solutions of the linearized Richards equation with arbitrary boundary and initial conditions: flux and soil moisture respectively

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Dominic Edwards
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1 Hydrology ays Soluons of h lnard Rchards uaon wh arbrary boundary and nal condons: flux and sol mosur rspcvly M. Mnan S. Pugnagh Unvrsà dgl Sud d Modna Rggo Emla p. Inggnra d Maral dllambn Va Vgnols 95 I- Modna Ialy E. Romano Unvrsà dgl Sud d Mlano parmno d Scn dlla Trra Son d Gofsca Va Ccognara 7 I-9 Mlano Ialy S. Vncn ISMAR - Isuo d Scn Marn namca Grand Mass CNR S. Polo 36 I- 35 Vna Ialy Absrac. Analycal soluons of dffrnal uaons dscrb physcal problms and provd gnral nsgh of h sudd naural mchansms. Alhough hy may b no suabl o solv complx hydrologcal problms hy ar fas and usful o s numrcal procdurs. Th soluons proposd n hs work ar oband for arbrary flux boundary condons and arbrary sol mosur nal condons. Ths prms o us sandard morologcal daa: prcpaon daa ncomng flux and Bown rao daa ougong flux whch ar vry common whl sol volumrc war conn masurmns ar usually no avalabl xacly a h sol-amosphr nrfac. A frs class of soluons s oband wh a unform nal condon for h sol mosur and a m dpndn surfac flux whch wll rprsns xprmnal prcpaon/ vaporaon cass. A soluon wh a mor gnral boundary condon s drvd usng a sum of smpl soluons oband for consan boundary condons. Fnally h sam chnu s appld o h sol mosur nal condon oo. Th vrcal profls of h sol war conn compud by hs smpl sum of soluons ar compard wh h rsuls of h aformnond analycal soluons.. Inroducon I s wll rcognd ha analycal soluons of dffrnal uaons dscrbng physcal problms provd gnral nsghs and concsly dnfy h rlaonshps among h varabls of h sudd problms allowng raonal approxmaons and smplfcaons. Thrfor alhough numrcal mhods ar powrful n solvng complx non lnar problms analycal soluons consrv hr uly and can also provd a usful chck o numrcal procdurs. Exac and approxmad analycal soluons of h non lnar dffrnal uaon govrnng h war flow n unsaurad sols Rchards uaon hav bn drvd by Sandrs al. 988 Hogarh al Unvrsà dgl Sud d Modna Rggo Emla parmno Inggnra Maral Ambn Va Vgnols 95 I- Modna Ialy mnan.marlna@unmo. Hydrology ays

2 Soluon of lnard Rchards uaon Parlang al. 99 Ross and Parlang 99 Parlang al. 997 Hogarh and Parlang among ohrs. Morovr analycal soluons of h lnard Rchards uaon hav bn drvd n ngral form by Warrck 975 and Basha 999 bu hr rsuls gv closd form soluons only for consan flux. Chn al. drvd analycal soluons of h lnard Rchards uaon for a vary of m dpndn fluxs bfor surfac sauraon. A dffrn approach o oban analycal soluons of h lnard Rchards uaon was uld by Mnan al. n prss assumng arbrary nal and boundary condons for h war conn. Th voluon of h las uod sudy s hr prsnd. Frsly a class of analycal soluons s oband assumng a unform nal condon and a known m dpndn flux a h surfac whch wll rprsns xprmnal prcpaon/vaporaon cass. Scondly a nw soluon s oband for any surfac flux boundary condon and any sol war conn nal condon. Ths s h rsul of h sum of smpl soluons oband for consan complmnary condons. Th vrcal profls of h sol war conn compud by hs smpl sum of soluons ar compard wh h rsuls of h aformnond class of analycal soluons. Th m bhavour of h ncomng flux a h surfac dscrbs ranfall nflraon or sprnkl rrgaon whos nnsy ar lowr hn h nflraon capacy of h sol. Morovr prcpaon masurmns ar much mor common hn sol mosur masurmns so soluons of h flow uaon oband assumng a m dpndn surfac flux boundary condon can b vry usful n suaons of nrs n hydrology.. Thory Consdr h lnard Rchards uaon sasfd by h sol war conn rangng from o : wh h followng arbrary condons: ; ; whr s h hydraulc dffusvy and k ; hy ar assumd consan. k s h sol hydraulc conducvy. On h bass of h work dscrbd n Mnan al. n prss h spac and m voluon of h sol war conn s gvn by h sum of wo soluons: whr: 9

3 Mnan al. [ ] 3 d d π π 3 Th soluon drvs from h nal condon of h problm and a null boundary condon whl h soluon drvs from h boundary condon of h problm and an nal condon ual o ro. In parcular f h soluon concds wh. L us assum now agan h nal condon for bu h boundary condon for h flux dfnd as:. Takng no accoun h lnary of h flux rlaonshp and of h dffrnal uaon h followng uaon wh nsad of as unknown can b wrn: Thrfor gvn h condons: ; corrspondng o: d d ; ; 5 for h flux h followng soluon s oband whr: [ ] 3 d d π π 6 Fnally wh smpl consdraons and rmmbrng ha lm h unknown funcon s oband: 5

4 Soluon of lnard Rchards uaon d 7 A pror uaon 7 gvs h dsrbuon of for any and bu h ngrals n uaons 6 and 7 may b dffcul or mpossbl o b solvd analycally. 3. Soluon wh xponnal flux a h surfac Snc ran gaug daa ar wdly collcd follows ha durng prcpaon vns h ncomng war flux rnd s known n many placs whl hs s no h cas of h sol volumrc conn. urng h las yars srong flood vns happnd boh n morologcally wllprdcd suaons and also n unprdcd local summr sorms always mor frunly. In any cas h floods ar srongly rlad o h sa of h sol mosur Obld and rboua. Lookng a rcn prcpaon vns can b sn ha many solad local summr cass can b dscrbd wh a surfac flux rprsnd wh h sum of fw xponnal funcons. Fg. shows four xampls corrspondng o a flux gvn by: a smpl xponnal funcon curv ; h sum of wo xponnals wh null prcpaon a curv ; 3 h sum of wo xponnals wh prcpaon no null a curv 3; h sum of hr xponnals curv corrspondng o a null prcpaon a and wh a null drvav a. Th ngral from ro o nfny of all h four curvs gvs a oal prcpaon of 5 mm. Th paramrs of h four uod funcons ar rpord n abl. Fgur. War Flux a h surfac Exampls of Boundary Condons 5

5 Mnan al. 3 3 N mm s - s - mm s - s - 3 mm s - 3 s Tabl. Paramrs of h four funcons rpord n fgur. Th soluon of assumng a null unform nal condon for h sol war conn and a flux boundary condon gvn by: s: rfc rfc rfc 8 Whr:. Clarly uaon 8 s vald only f. If > h soluon of s mor complca and nvolvs h rror funcon of complx argumn Abramow and Sgun 965. u o lnary of usng a boundary condon whch s h sum of wo or hr xponnal funcons h soluon s gvn by h sum of wo or hr soluons lk 8. Fg. 3 shows sold lns h rnd of h vrcal profl of h sol volumrc war conn oband assumng a unform nal condon and as boundary condon. In hs horcal xampl fv vrcal profls of h sol volumrc war conn can b sn. Th hn vrcal sold ln s h nal condon whl h ohr sold curvs ar oband wh a consan m sp. Th war conn profl rnd shows h m ncras of h sol mosur as h war nrs h sol. Th las wo curvs show an nflcon pon movng downwards whl hs s no vdn n h frs wo. Th crcls dscrb h soluon oband usng h approxmang funcon prsnd n h followng paragraph.. Soluon wh sp funcons approxmang h flux a h surfac No always h xprmnal surfac flux can b rprsnd by a smpl funcon. In such a cas may b vry dffcul or vn mpossbl 5

6 Soluon of lnard Rchards uaon o solv h ngral n uaons 6 and 7. Vc vrsa h soluon of uaon s smpl for h followng complmnary condons: ; ; 9 whr s a consan; ha s a unform ro nal condon for h sol volumrc war conn and a consan flux a h surfac. From uaons 6 and 7 h soluon of uaon rsuls: rfc Irfc rfc Irfc s h rad complmnary rror funcon. Approxmang any arbrary boundary condon wh a sum of sp funcons h soluon of h problm s gvn by h sum of xprssons smlar o uaon. In fac assumng a unform nal condon.. and a boundary condon as skchd n Fg. h soluon n h m nrval M- M rsuls: M rfc Irfc rf M s h numbr of dsconnus a... M... whr h boundary condon assums h valus:... M... bsds. Fgur. Sp funcons usd o approxma h surfac war flux. In Fg. a consan flux lass from o a consan flux lass from and and so on. In uaon h dffrnc - rprsns h hgh of h sp funcon sarng a - clarly. 53

7 Mnan al. Fgur 3. Sol volumrc war conn profls oband usng uaon 8 sold lns and h approxmad profls from uaon. In Fg. 3 h sold lns ar h rsuls oband from uaon 8 accordng o h dscrpon prvously don; hy ar compard wh h approxmad soluon crcls. Th sol mosur rsuls agr sasfacory vn hough a raw m rsoluon was usd o approxma h ncomng flux. In ral xprmnal cass hs chnu can b usd o choos a propr ran gaug acuson m. 5. Soluon wh sp funcons approxmang h nal condon In h prvous paragraph h boundary condon.. h flux a h ar-sol nrfac was approxmad by a sum of sp funcons. In a smlar way hr an arbrary nal condon s approxmad by: N H n n n n N s h oal numbr of dsconnus a... N whr h nal condon assums h valus:... N bsds. Hx s h Havsd funcon wh argumn x Jons 966. Now assumng a null flux as boundary condon h soluon of uaon says ha h sol volumrc war conn s gvn by h sum of N soluons: N n n 3 whr from uaon 6 and 7 on obans: 5

8 Soluon of lnard Rchards uaon rfc rfc Irfc and n a smlar way bu wh som mor dffculs on obans: n n n n rfc n n rfc Irfc 5 Gnrally approxmang any arbrary boundary and nal condon wh a srs of sp funcons h soluon of h Rchards uaon s gvn by h sum of uaons and Concluson In hs work h lnard Rchards uaon wh a boundary condon on h flux and a sol war conn nal condon has bn solvd n ngral form. A class of closd form analycal soluons has bn drvd for a flux boundary condon whch s h sum of xponnal funcons. On h ohr hand many ral prcpaon vns may b rprsnd as h sum of fw xponnal funcons. A mor gnral soluon s oband approxmang h flux boundary condon by h sum of sp funcons and wh a null unform sol war conn nal condon. For hs soluon h mahmacal consran dscussd n paragraph 3 dosn xs. Fnally a soluon s oband for a null flux a h surfac and an nal condon approxmad by h sum of sp funcons. Th xprsson oband addng h las wo soluons prms o solv h lnard Rchard uaon for any arbrary boundary and nal condon. Th xprsson oband from h dscrbd procdur s no xacly an analycal soluon bu can b vry usful o solv hydrologcal problms. In parcular h procdur allows usng xprmnal ran gaug daa whch ar vry common. In fac hs daa may b assumd as h ncomng war flux a h amosphr-sol nrfac f h prcpaon ra dosn xcd h sol nflrably. Rfrncs Abramow M. Sgun I.A. 965: Handbook of mahmacal funcons. ovr Publcaon Nw York Basha H.A. 999: Muldmnsonal lnard nonsady nflraon wh prscrbd boundary condons a h sol surfac. War Rsour. Rs Chn Jann-Mou Tan Yh_Ch Chn Chu_Hu Parlang J.Y. : Analycal soluons for lnars Rchards uaon wh arbrary m-dpndn surfac fluxs. War Rsour. Rs

9 Mnan al. Hogarh W.L. Parlang J.Y. : Applcaon and mprovmn of a rcn approxma analycal soluon of Rchards uaon. War Rsour. Rs Hogarh W.L. Parlang J.Y. Braddock R.. 989: Frs ngrals of h nflraon uaon Nonlnar conducvy Sol Sc Hogarh W.L. Parlang J.Y. Norbury J. 99: Addndum o Frs ngrals of h nflraon uaon Sol Sc Jons. S. 966: Gnralsd funcons. McGraw-Hll Nw York Mnan M. Pugnagh S. Vncn S. Sananglo R. n prss: War Mass Balanc n h Surfac Sol: Som Parcular Analycal Soluons of h Flow Euaon and h Exprmnal Masurmns of h Alpn Toc Vally Cas Sudy. Clma and Hydrology n Mounan Aras Edd by C. d Jong. Collns and R. Ran John Wly & Sons n prss. Obld Ch. and rboua A. : Quanav prcpaon forcass: a ral m xrcs durng h MAP xprmn. Hydrologcal Aspcs n h Msoscal Alpn Programm SOP Exprmn Edd by B. Bacch and R. Ran Tchncal Rpor of h Unvrsy of Brsca p. of Cvl Engnrng N..VII. hp://cvsrv.ng.unbs./un/ran/map/sop/r7lh.pf Parlang M.B. Prasad S.N. Parlang J.Y. Romkns M.J.M. 99: Exnson of h Hasl-Alksn chnu o arbrary sol war dffusvs. War Rsour. Rs Parlang J.Y. Barry.A. Parlang M.B. Hogarh W.L. Havrkamp R. Ross P.J. Lng L. Snhus T.S. 997: Nw approxma analycal chnu o solv Rchards uaon for arbrary surfac boundary condons. War Rsour. Rs Ross P.J. Parlang J.Y. 99: Comparng xac and numrcal soluons of Rchards uaon for on-dmnsonal nflraon and dranag. Sol Sc Sandr G.C. Parlang J.Y. uhnl V. Hogarh W.L. Lockngon. O an J.P.J. 988: Exac nonlnar soluon for consan flux nflraon. J. Hydrol Warrck A.W. 975: Analycal soluons o h on-dmnsonal lnard mosur flow uaon for arbrary npu. Sol Sc

Consider a system of 2 simultaneous first order linear equations

Consider a system of 2 simultaneous first order linear equations Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm