New Mexico Tech Hyd 510

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1 New Meo eh Hy 5 Hyrology Program Quanave Mehos n Hyrology Noe ha for he sep hange problem,.5, for >. he sep smears over me an, unlke he ffuson problem, he onenraon a he orgn hanges. I s no a bounary onon. ransform Soluons o Hea ffuson see Crank,. I an be shown by mensonal analyss ha soluons o 3 are ofen of he form A, Φ 68 / where Φ s a funon o be eermne 9. In parular, we ve alreay foun Gaussan an error funon soluons. Noe ha he funon s self-smlar. ha s, he funon s srehe one way or anoher, bu he shape remans unhange over me-spae. You ve seen suh smlary soluons before, n he hes Well funon, whh has smlary varable ur S/, where he parameers ake on he normal well-hyrauls efnons, an hyraul ffusvy s /S. I s no unusual o assume a soluon of he form of 68 an hen o seek he oeffen A, ha mahes he bounary an nal onons. As a emonsraon, le s ake a look a one suh applaon of smlary o fn anoher pah o a famlar soluon. Bolzmann ransformaon Crank, 975, p. 5; hs page s no nlue n he supplemenal reang. Reonser he ffuson equaon gven by 69 wh nal an bounary onons for a sem-nfne oman esrbe by IC,, a onsan BC,, a onsan 79a,b, BC,, a onsan From our prevous work we mgh epe ha he soluon nvolves an error funon. Insea, le s assume only ha s a smlary soluon of he ype shown n 68. efne a smlary varable 7 hen apply hs hange n varables an some han rules o onver 69 no an OE. Converng a PE o an OE s he goal of many mehos ha solve parabol PEs. In hs ase he han rules are 9 ypally s some ombnaon of eponenal or error funons. -99-

2 New Meo eh Hy 5 Hyrology Program Quanave Mehos n Hyrology -- / / / 7a,b, Noe he ornary ervaves wh respe o he smlary varable,. We an subsue no he ffuson equaon, / / or, hs smplfes o + or + 73 whh s a seon orer OE. We an solve by reuon n orer, say be efnng v 75 where he symbol v was hosen arbrarly. he OE beomes s orer, + v v 76 Is soluon, by separaon of varables SOV an negraon, s v v ln v + a ep v ep a 77 where an a are onsans, an we have erve a new s orer OE o solve. We an apply SOV agan o ge

3 New Meo eh Hy 5 Hyrology Program Quanave Mehos n Hyrology a ep 78 Obvously, we shoul negrae hs epresson. However, o progress furher we nee o apply nal an bounary onons. Reall ha s hese onons ha eermne he form of he soluon o he PE. In fa, wh he Bolzmann approah, s only when he nal an bounary onons are epressble n erms of alone, an an are no nvolve separaely, ha he Bolzmann ransformaon an be use Crank, 975, p. 6. For eample, n our ase, when, no maer he value of, hen. Smlarly, when, hen. Wha abou when? We have, agan. hus our onons beome IC & BC, a onsan BC, a onsan 79a.b Le s use he seon onon an negrae from o /, a ep a erf 8 where we an reognze he negral as appearng n he error funon. ha leaves one onsan o eermne, from he oher onon,, where erf erf, or a erf a 8 / Subsung bak no 8 gves he fnal soluon of he PE, erf 8a Flu alulaon a : One applaon of hs soluon s o he alulaon of he me varyng flu of hea no he oman from he bounary, or o he alulaon of he oal hea ae. he frs of hese nvolves Fourer s law an akng he ervave of 8 wr, whle he seon nvolves he negral of ha ervave over me, or of 8 over spae. he flu s gven by, + erf 8b ffusve flu -K K erf 83 where K s he hea onuvy, an K/ρ. he ervave on he rgh evaluaes usng he han rule p. of hese noes an by applyng he ervave of an negral wh a varable upper lm.e., he ervave of an error funon; p. of hese noes. --

4 New Meo eh Hy 5 Hyrology Program Quanave Mehos n Hyrology [ erf ] ep [ erf ] ep hus, he ffusve flu any, s Flu K K ep 8 85 A he bounary, he flu no or ou of he oman s gven by applyng hs moel a, or Bounary ffusve flu a K K Kρ 86 he bounary flu s greaer for hgher hermal onuvy an hea apay, larger emperaure fferenes, an smaller me. I ereases wh he square roo of me. We negrae hs over me o ge he quany of hea ae as a funon of me, H, or H / Kρ Kρ Kρ 87 / he amoun of hea ae nreases wh he square roo of me. he amoun H shoul also equal he spaal negral of hea ae, or H ρ ρ [, ] ρ Kρ [ erf ] ρ [ erf ] erf 88 whh agrees wh 87. I ha o look up he negral a hp://mahworl.wolfram.om/erf.hml, he mahemaa web se, o fn ha erf 89 In applaon, f I ha no been able o fn hs negral, I woul have rele on 87. I s ne o have more han one way o solve a problem. --

5 New Meo eh Hy 5 Hyrology Program Quanave Mehos n Hyrology Oher applaons We an apply hs moel elsewhere n hyrology. One ommon applaon s bank sorage ue o a passng floo wave. If h s he nal hea n he aqufer, h s he hegh of he floo wave usng he same aum, s hyraul ffusvy /S y, hen by homology o 8, 86 an 87 he aqufer hea, he bounary flu no he aqufer, an he volume of waer sore are gven by h, h + h h erf h h h h h S y 9a,b, V S y h h hs ells us ha he rae of bank nflraon an he amoun of bank sorage s greaer for hgher ransmssvy an sorage oeffen, an larger hea fferenes h h. he rae ereases wh he square roo of me, whle he sorage volume nreases wh he square roo of me. Anoher ommon applaon s mosure absorpon no a rer or esorpon from a weer porous maeral ue o apllary an gnorng gravy, prove we assume ha apllary ffuson s a lnear proess. If θ s he nal mosure onen n he sol volume of waer per volume of sol, θ s bounary mosure onen, s apllary sol mosure ffusvy K ψ/θ, where ψ s sol mosure enson an K s hyraul onuvy, hen by homology o 8, 86 an 87, he mosure onen, bounary flu no he sol, an he oal amoun of absorpon are gven by θ, θ + θ θ erf θ θ θ θ θ V θ θ 9a,b, he rae an amoun of mosure absorpon s greaer for hgher sol mosure ffusvy an larger mosure fferenes θ θ. he rae ereases an he amoun nreases wh he square roo of me. If we a gravy o hs mosure problem we an suy nflraon, bu onnung o assume ha K an are onsans an he problem lnear. In hs ase only he flu hanges, or θ θ θ + K + K 9 hus he presene of gravy nreases he nflraon rae, bu he rae s suppresse by hgher nal mosure, θ. he aual problem s more omplae han hs. Boh an K epen on Assumng ha s posve ownwar. -3-

6 New Meo eh Hy 5 Hyrology Program Quanave Mehos n Hyrology mosure onen. he problem s aually non-lnear. Le s reurn o he hea ffuson problem o eplore hs ssue. Non-lnear ffuson Reonser he ffuson equaon bu for he ase of a ffuson oeffen ha epens on he sae. For hea ffuson hs woul be gven by 93 Oher applaons of sae epenen ffuson oeffen nlue solue ffuson, grounwaer hyraul ffuson where n a phrea aqufer he ransmssvy epens on he saurae hkness an hus on he waer able elevaon, an mosure absorpon an nflraon. How an we solve 68 for any general funon, whou spefyng ha funon eplly? We ll see ha wha we an o s o evelop a soluon approah, mplemen an arry ou he symbol alulaons o a eran pon, bu hen we ll have o apply a parular funon an omplee he remaner of he alulaon numerally. Below we assume ha epens on emperaure hrough a epenene of hermal onuvy K on emperaure. Hea apay s assume onsan. Whle here are some very goo non-lnear soluon approahes for 68, also apable of hanlng aveon, we ll fous on he smples. I s o llusrae only; you won be applyng hs meho yourself. Basally, he ea s o use he Bolzmann ransformaon agan. We assume ha he soluon wll be somehng lke ha n he smlary soluon of 68, apply he hange of varables n 7 bu n erms of a fe aken a he bounary, an erve a non-lnear OE smlar o 73, or hs s presene n some eal n Crank 975, Chap. 7, an revewe n he one of sol mosure absorpon an nflraon n Eagleson ynam Hyrology, Prene Hall, 97, pp For eample, he bounary ffusve flu a beomes, K Kρ 86 John Phllp CSIRO an Yves Parlange Cornell are well known n hyrology, sol physs, an mahemaal physs for her onrbuons along hese lnes. For eample, boh worke on somehng alle he flu onenraon approah. --

7 New Meo eh Hy 5 Hyrology Program Quanave Mehos n Hyrology whh, by omparson o he lnear ase, we an rewre as K Kρ 87 where he weghe mean onuvy s gven by K K 88 an K s aken a he bounary, K K ρ. Crank Conuon of Hea n Sols, Ofor Press, 959 some numeral epermens o eermne appromaons for K, an hen eplore he soluon for varous funons K. You an also fn some of hs n Eagleson 97. For eample, Crank foun ha as long as K an nrease wh hen he bes weghng for nreasng emperaure ases s gven by 5 3 5/ 3 / K K 89 3 One woul ake epermenal aa on K, perform hs negraon perhaps numerally, an nser no 87 o ge he moel for hea flu. Smlar alulaons are one for emperaure srbuon an oal onen. Homologous alulaons are one for sream bank sorage an mosure absorpon an nflraon. -5-