Long-term morphodynamic modelling of tidal basins with rivers

Chap e 4: Linea app oxima ion o ASMITA equa ions 27 4 Linea app oxima ion o ASMITA equa ions 4.1 In oduc ion In his chap e , a simpli ied app oxima ion o ASMITA equa ions will be done ying o ob ain mo e insigh in o he mo phological beha iou o idal inle s when a i e discha ge is conside ed. Fo his, Taylo se ies a e used and he app oxima ions a e done a ound he equilib ium olume o each elemen . Thanks o hese simpli ica ions, we can ob ain exp essions which le us know how olumes e ol e wi h ime. We will s a wi h a idal sys em cons i u ed by one elemen , and inc ease he di icul y o he sys em, adding elemen s one by one, un il a sys em wi h h ee elemen s. Finally, new ASMITA equa ions will be p esen ed in he case o a n-elemen s sys em. 4.2 One elemen in he idal sys em 4.2.1 Model equa ions Fi s , a sys em wi h only one elemen will be conside ed. As we also may ake in o accoun he i e e ec , his single elemen migh be he channel, o he wise he i e e ec would no be e lec ed p ope ly. We will p esen a igu e whe e exchange sedimen ela ions be ween elemen s is isible. Q Q Q Qw ww w Qc QcQc Qcc cc c Q Q Q Qw ww w () oc c E cc δ − () cc c ce wA c c− ou side wo ld S SS S channel we olume d y olume Figu e 4-1 Sedimen exchange in case o one elemen in he sys em A e his epo Chap e 4: Linea app oxima ion o ASMITA equa ions 28 In his case, model equa ions can be w i en as i ollows: () () cE c sccec cc QcSwAc c δ −+ −= − (4.1) () c s cce c dV wA c c d =− (4.2) ce ce E c V cc V æö =ç÷ èø (4.3) F om he i s equa ion, which co esponds o he mass-balance equa ion, we can de e mine c c: ce EscE c c sc V cSwAc V cQwA δ δ æö ++ ç÷ èø =++ (4.4) We will subs i u e c c and ce c in o he equa ion ha ep esen s mo phological changes (4.2) and we ob ain: ce EscE c cce sc E csc V cSwAc V dV V wA c d V Q w A δ δ éù æö ++ êú ç÷ æö êú èø =− ç÷ êú ++ èø êú êú ëû (4.5) Now, he only a iable ha is unknown is he ac ual olume o he channel: c V. We will w i e again he p e ious equa ion in ano he way: () () csc ce EE sc c dV w A V cQ cS d Q w A V δδ δ ìü æö ïï =+−+ íý ç÷ ++ èø ïï îþ (4.6) And i can be schema ise as i ollows: c c dV a AB C d V ìü æö ïï =− íý ç÷ èø ïï îþ (4.7) Chap e 4: Linea app oxima ion o ASMITA equa ions 29 F om his, i becomes clea ha he equa ion abo e is a non-linea di e en ial equa ion. In o de o sol e i , we will linea ise his equa ion, using Taylo se ies app oxima ions, a ound he equilib ium olume Vce, which co esponds o he cons an a in he equa ion (4.7). The second and highe o de e ms will be neglec ed. We will change ou a iable c V o ' c V, as i ollows: ' ccce VVV=− (4.8) And, a e linea ising equa ion (4.7) we will ob ain a linea non- homogeneous di e en ial i s o de equa ion: () '''' ' c cE c dV BA ABV AcQS V d a =− = −− (4.9) Whe e: () 'sc E s c wA cQ S AQwA δ − =++ (4.10) () '() scE sc w A c Q BQwAa δ δ + =++ (4.11) Nex , he solu ion o he p e ious di e en ial equa ion is p esen ed: '(') co VAVAe τ ττ − =+ − (4.12) Whe e : () () ce s c scE VQwA w A c Q δ τδ ++ =+ (4.13) sc s c wA AQwA δ =++ ; (4.14) () () Ece E cQ SV A c Q τδ − =+ (4.15) () ' , and 0 ooce o VVV VV =− = =, his is he ini ial condi ion necessa y o sol e he di e en ial equa ion Chap e 4: Linea app oxima ion o ASMITA equa ions 30 F om equa ion (4.12) we can see ha o →∞, a s eady s a e is eached, as ' c V end o an asymp o ic alue. The co esponding channel olume is: ,1() E cce E cQ S VV c Q δ ∞ æö − =+ ⋅ ç÷ + èø (4.16) This olume, hence o h called he end olume, de ia es om he equilib ium olume ce V as i is clea ly exp essed in las equa ion. We migh keep in mind ha ha ce V co esponds o an inle sys em unde he in luence o a ide only. Wi h he new a iable we ha e in oduced, he i e , he sys em needs o be anspo ing sedimen con inuously om he channel o he ou side wo ld, acco ding o he endency o each an equilib ium si ua ion; o he wise he channel would disappea , because o he ex a amoun o sand accumula ed. Despi e o ha , we migh no o ge ha he i e means also a wa e discha ge o he channel. As explained be o e, he anspo exis s when he sys em is no in equilib ium, hen, in ou case, he channel needs o be all he ime in an "ou o equilib ium" s a e so he anspo is able o occu cons an ly. 4.2.2 Time-scales One can in e p e a ime-scale as he cha ac e is ic ime o adap a ion, once an elemen is mo ed om i s equilib ium, and i migh be di e en o each elemen . Also, his ime-scale le us know abou he i s esponse o an elemen in on o a dis u bance. Acco ding o Eysink (1990), one can de ine he ime-scale as i ollows (in case o no i e in he sys em): 0 ' ' o V dV d τ = =− (4.17) The nomina o can be in e p e ed as he ini ial olume and he denomina o as he ini ial a e o adap a ion. Also, his can be seen g aphically as: Chap e 4: Linea app oxima ion o ASMITA equa ions 31 i me τ o V ()V As one can obse e in he p e ious g aph, he ime scale τ co esponds o he in e sec ion be ween he angen a =0 and he ime axis. Fu he mo e, we can ead he p e ious equa ion as: 0 '' o dV V d τ = =− (4.18) I can be concluded ha as la ge is he slope o he angen line, which means 0 ' dV d = bigge , as lowe is he alue o τ ( ' o V does no change). Also, low alues o ime scale mean ha in less ime he sys em will each a s eady si ua ion. Le 's w i e again he p e ious equa ion ha desc ibes olume e olu ion in he channel in case o a i e discha ge: '(') co VAVAe τ ττ − =+ − (4.19) We will change A τ o ,' c V∞, hus: ,, ''(' ') ccoc VV VVe τ − ∞∞ =+− (4.20) Figu e 4-2 Time-scales g aphical in e p e a ion A e his epo Chap e 4: Linea app oxima ion o ASMITA equa ions 32 In his case, jus de i a ing p e ious exp ession, he a e o change is gi en by: , '' ' cc cVV dV e d τ τ − ∞ − éù =−êú ëû (4.21) Acco ding o Eysink (1990), he ime scale pa ame e can be exp essed as i ollows: , 0 '' ' cc c VV dV d τ ∞ = − = (4.22) No ice ha las exp ession can be ob ained, oo, om equa ion (4.21) when =0. We will y o unde s and he meaning o p e ious o mula. Fi s o all, we will pay a en ion o he nomina o : () () * ,,, '' cc cce c ce cc c VV VV V V VV V ∞∞∞ −=−−−=−= (4.23) Hence, his is he di e ence be ween he end olume ,c V∞ and he equilib ium olume ce V. Then, * c V is a ela i e olume, bu compa ed o he end olume ins ead o he equilib ium olume. The e m τ ( he ime scale) will be he in e sec ion be ween he angen o he cu e '( ) c V when =0 and he ho izon al s aigh line , '' cc VV ∞ =. Thus: * * 0 o V dV d τ = =− and *'dV dV d d = (4.24) because hey only di e om a cons an which disappea once we de i e. Nex g aphs help o unde s and p e ious explana ions: Chap e 4: Linea app oxima ion o ASMITA equa ions 33 '( ) c V , 'c V∞ τ ime Figu e 4-3 Time-scales using olumes ela i e o Veq A e his epo *() c V ,c V∞ τ ime Figu e 4-4 Time-scales using olumes ela i e o Vend A e his epo In he exp ession o ime scales, in bo h si ua ions: wi h and wi hou i e , he e m τ, he so called ime-scale, ma ch up wi h he e m ha appea s in he powe o he exponen ial unc ion ( ha desc ibes he elemen beha iou wi h ime). In he nex chap e s, when we conside mo e han one elemen in he idal sys em, we will see ha he e m τ, de ined as in he equa ion (4.17) and he e m ha appea s in he powe o he exponen ial unc ion a e no he same. Howe e , his will be explained mo e clea ly when exp essions o a sys em wi h mo e han one elemen a e w i en. Chap e 4: Linea app oxima ion o ASMITA equa ions 34 4.2.3 Applica ion o he linea ised equa ions Le 's conside an example o isualise he beha iou o he sys em. We will also compa e his si ua ion wi h a si ua ion wi hou i e discha ge (keeping he es o he inpu s wi h he same alue). Be o e ha , we will p esen some cla i ica ions o make clea he name used o he di e en a iables. The e m ela i e olume co espond o 'V, hence, is he olume compa ed o he equilib ium olume, in o he wo ds, i shows how much a olume di e s om he equilib ium olume (de ined in a no i e discha ge si ua ion). The e m end olume is he olume a he s eady si ua ion eached by he sys em when enough ime has passed. I co esponds o he non- homogeneous e m in he equa ion (4.20), and i is he e m A τ in equa ion (4.19), o ins ance. Thus, i is he limi when ime end o in ini e, and i is equal also o he olume ha makes he de i a i e 'dV d become ze o, as he e is an asymp o ic cha ac e in he olumes e olu ion. He e we p esen he da a used as an example. These da a’s a e ob ained based on Van Goo (2001) wo k, and we ha e included an inexis en i e discha ge: cE ws n δoc Ac h Vo Q S 0.0002 5e-5 m/s 2 1500m3/s 9.83e7 m3 2 m a iable 200 m3/s 0.2 m3/s Table 1 Inpu da a o one elemen sys em A e Van Goo (2001) Acco ding o Ma in, J.P (1997), he sedimen inpu is been es ima ed as app oxima ely 1/1000 x Q. Nex , a g aph ha shows how he olume o he channel change wi h ime in case o a i e discha ge in he basin is p esen ed, using p e ious inpu da a: Chap e 4: Linea app oxima ion o ASMITA equa ions 35 Q=200m3/s, S=0,2m3/s -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 1,5E+08 0 50 100 150 200 250 300 350 (yea s) we ela i e olume (m3) Vo=2e8 Vo=4e8 Vo=6e8 Vo=1e8 Clea ly, one can see he asymp o ic endency o he elemen since i e ol es owa ds a s eady s a e, which do no co espond o he o iginal equilib ium olume. In he ollowing g aph we will see he beha iou o he channel when he e is no i e discha ge. I will e ol e o he own equilib ium olume, o a ela i e olume equal o ze o: Q=0, S=0 -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 1,5E+08 2,0E+08 0 50 100 150 200 250 300 350 (yea s) we ela i e olume (m3) Vo=2e8 Vo=4e8 Vo=6e8 Vo=1e8 Figu e 4-5 Channel olume e olu ion wi h i e discha ge A e his e p o Figu e 4-6 Channel olume e olu ion wi hou i e discha ge A e his epo Chap e 4: Linea app oxima ion o ASMITA equa ions 42 The cons an s C1 and C2 may show us how he elemen s e ol e, ela ed o he (sys em) ime-scales T1 and T2. We shall no o ge ha hese cons an s depend on he ini ial condi ions and ha ini ial condi ions do no in luence he end olumes in each elemen . F om nex sec ion, whe e we apply he linea ised equa ions o an example, we ake he exp ession (4.48) and lea e C1 and C2, hus: 8 36,1 15,3 12 7 '1 1 1,34 10 '0,38 1,29 3,08 10 c d VCeCe V −− æö −⋅ æö æö æö =++ ç÷ ç÷ ç÷ ç÷ −⋅ èø èø èø èø (4.47) To see mo e clea ly which is he in luence o each cons an , we will gi e he alue equal o he uni o one o hem, and obse e how he o he one e ol e when we change i s alue. In he ollowing g aphs we can see i s he e ec o keeping C1 equal o he uni and changing C2, and second he e ec o changing C1 while C2 emains cons an and equal o he uni : -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 0 20406080100 (yea s) V' (m3) Figu e 4-9 In luence o C1 in eg a ion cons an A e K ag wijk (2001) C1 a iable, C2 =1 C CC C1 11 1>0 >0>0 >0 C CC C1 11 1<0 <0<0 <0 C CC C1 11 1<0 <0<0 <0 C CC C1 11 1>0 >0>0 >0 channel del a Chap e 4: Linea app oxima ion o ASMITA equa ions 43 -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 0 20 40 60 80 100 (yea s) V' (m3) F om hese igu es, one can obse e ha he ini ial de elopmen is de e mined by he smalles ime-scale which co espond o T2, his becomes clea in he i s igu e. A e a while, he esponse o he smalles ime-scale disappea s, and wha emains is he de elopmen o he la ges ime-scale. One can see ha obse ing he second igu e. Also, depending on he sign o he cons an , we can see an inc easing o dec easing beha iou . 4.3.3 Applica ion o he linea ised equa ions In o de o isualise he beha iou o he elemen s, we will conside some da a and used he linea ised equa ion jus sol ed. Nex , he da a a e p esen ed: cE wsi n δij Ai h Voi Q S Channel 0.0002 1e-5 m/s 2 1500m3/s 7.47e7 m22 m 2e7 m3 200 m3/s 0.2 m3/s Del a 0.0002 5e-5 m/s 2 1500m3/s 9.83e7 m2 2 m 5e7 m3 200 m3/s 0.2 m3/s Table 2 Inpu da a o a wo elemen s sys em A e Van Goo (2001) C1 =1, C2 a iable Figu e 4-10 In luence o C2 in eg a ion cons an A e K a g wi j k (2001) C CC C2 22 2>0 >0>0 >0 C CC C2 22 2<0 <0<0 <0 C CC C1 11 1<0 <0<0 <0 C CC C1 11 1>0 >0>0 >0 channel del a Chap e 4: Linea app oxima ion o ASMITA equa ions 44 The esul s ob ained in his example a e: 8 77 36,1 15,3 7 '1 1 1,34 10 4,63 10 6,0 10 '0,38 1,29 3,08 10 c d Vee V −− æö −⋅ æö æö æö =⋅ +⋅ + ç÷ ç÷ ç÷ ç÷ −⋅ èø èø èø èø (4.48) And he co esponding g aph: Volumes -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 0 50 100 150 200 250 300 350 ime (yea s) elemen olume (m3) channel del a F om he g aph we con i m he endency o he elemen s o e ol e o a s eady s a e. As in he case o one elemen , he sys em needs o be anspo ing he sedimen om he i e o he sea, wha means ha he e will be ho izon al anspo con inuously om he channel o he del a, and om he del a o he ou side wo ld bounda y. A e wa ds, we will obse e he beha iou o he idal inle wi hou i e discha ge, wi h Q=S=0. As one migh be expec ing, he sys em e ol es o an equilib ium si ua ion whe e olumes in each elemen a e equal o he equilib ium olume. Nex , he g aph ha co esponds o he same da a as he si ua ion below bu wi hou he i e discha ge is p esen ed: Figu e 4-11 Elemen 's olume e olu ion wi h i e discha ge A e his epo Chap e 4: Linea app oxima ion o ASMITA equa ions 45 Volumes -2,0E+08 -1,5E+08 -1,0E+08 -5,0E+07 0,0E+00 5,0E+07 1,0E+08 0 50 100 150 200 250 300 350 ime (yea s) elemen olume (m3) channel del a 4.3.4 Ini ial mo phological esponses o di e en dis u bances We know ha elemen s end o e ol e o an equilib ium s a e, hey exhibi an asymp o ic beha iou . This e olu ion is no necessa ily mono onous. The ini ial esponse o he elemen immedia ely a e a dis u bance can be mo e away om he end olume, bu also, he elemen can o e shoo he end olume. In he i s case, we alk abou an opposi e ini ial esponse (bump) and in he second case an o e shoo (K ag wijk, 2001). Nex igu e will helps us o unde s and hese wo ini ial esponses: Figu e 4-13 Ini ial esponse in on o a dis u bance A e K ag wijk (2001) a. o e shoo b. bump Figu e 4-12 Elemen 's e olu ion wi hou i e discha ge A e his epo Chap e 4: Linea app oxima ion o ASMITA equa ions 46 We will use he linea ised equa ions o wo elemen s o see whe he an "o e shoo " o a "bump" esponse will occu . Tha will depend on he in eg a ion cons an s and on he ini ial olumes. They bo h de e mine he cha ac e o he ini ial mo phological beha iou . To unde s and he di e en esponses he e can be, i is easie i we wo k wi h he ini ial ela i e olumes, bu ela i e o he end olumes ins ead o he equilib ium olumes. Anyway, we will also name hem ini ial ela i e olumes, bu we shall no o ge hey a e compa ed o he end olumes. We will s udy he sign o hese olumes (when =0) o he example p esen ed be o e, in chap e 3. Figu es showing his s udy will be p esen ed in Appendix 1. Fu he mo e, in ou example, we should no o ge ha C1 co esponds o he beha iou a long e m and C2 co esponds o he beha iou a sho e m (compa ed o he o he one), as he ime-scales show, acco ding o explana ions in sec ion 4.3.2. The p ocedu e ollowed is p esen ed nex . Fi s o all, we ew i e he equa ion (4.47) a =0: 8 12 7 '1 1 1,34 10 '0,381,29 3,08 10 co do VCC V æö −⋅ æö æöæö =++ ç÷ ç÷ ç÷ç÷ ç÷ −⋅ èøèø èø èø (4.49) whe e ''(0) ''(0) co c do d VV VV == == (4.50) And we will wo k wi h ini ial olumes compa ed o he end olumes, hence, wi h ela i e ini ial olumes. Thus: *8 *7 '''1,3410 '''3,0810 co co cend co do do dend do VVV V VVV V =− =+ ⋅ =− =− ⋅ (4.51) A e wa ds we will ob ain C1 and C2 as a unc ion o he ela i e ini ial olumes as i ollows: ** 1 ** 2 1, 29 1, 67 0,38 1, 67 co do co do VV C VV C − = + = (4.52) Chap e 4: Linea app oxima ion o ASMITA equa ions 47 Mo eo e , i is easy o see ha he ini ial ela i e olumes a e olumes compa ed o he end olume, which a he same ime is also a ela i e olume, since i is compa ed o he equilib ium olume. Bu , also, i is easy o see ha he ini ial ela i e olumes a e he di e ence be ween he ini ial ne olume and he end ne olume: ()( ) *'' io io iend io ieq iend ieq io iend VVV VV V V VV=− = − − − =− (4.53) The e o e, we can alk ei he abou di e ences be ween ini ial and end olumes; whe he hey a e ela i e o ne olumes do no a ec he ela i e ini ial olumes * io V. Knowing ha he esponse o he sys em may change depending on he sign o C1 and C2 (as i has been explained in p e ious sec ion), we will see o which alues o he ini ial ela i e olumes hose cons an s a e posi i e o nega i e. We will conside he plane () ** , co do VV and d aw on i he s aigh lines C1 =0 and C2 =0. Axes * co V and * do V di ide he plane in 4 zones wi h he 4 possible combina ions o signs. The same will occu wi h he cons an s C1 and C2. Conside ing bo h pa ame e s simul aneously (in eg a ion cons an s and ini ial ela i e olumes) we ob ain he plane di ided in 8 zones: -16 0 16 -10 0 10 Vco* Vdo* C1=0 C2=0 1 2 3 4 5 6 7 8 Figu e 4-14 Eigh zones depending on ini ial condi ions A e his epo As one can obse e, we ob ain 8 di e en a eas, depending on he sign o C1 and C2 and on he sign o he ini ial ela i e olumes. The sign o he olumes shows whe he he ini ial olume o he elemen is highe o Chap e 4: Linea app oxima ion o ASMITA equa ions 48 lowe han he end olume. Also, he sign o he cons an C2 shows he esponse o he elemen in a sho e m and he sign o he cons an C1 shows he esponse a long e m (as jus explained p e iously). Conside ing all possible combina ions be ween signs o C1 and C2 and ini ial ela i e olumes o he channel and he del a, we can de e mine he cha ac e o he ini ial mo phological beha iou . In o de o keep in mind he in luence o he cons an s o he beha iou o he sys em, we will p esen a igu e o he channel and o he del a: channel (yea s) olume (m3) c1>0 c1<0 c2>0 c2<0 Figu e 4-15 Ini ial condi ion cons an o he channel A e his epo del a (yea s) olume (m3) c1>0 c1<0 c2>0 c2<0 Figu e 4-16 Ini ial condi ion cons an o he del a A e his epo Chap e 4: Linea app oxima ion o ASMITA equa ions 49 Le 's conside a case whe e he del a has a posi i e ini ial olume, ha is o say an ini ial olume la ge han he end olume. Le 's conside C2 posi i e, hen, he e is an ini ial endency o dec ease in he elemen ( his becomes clea i we look a he igu e abo e, numbe 4-16. Le 's conside also ha C1 is posi i e, hen, a long e m, he e will be a endency o inc ease. The e o e, we will see ha he elemen o e shoo s he end olume a a sho e m (dec easing endency), and hen, ends o e ol e o his end olume a long e m ( he olume is inc easing). Conside ing all possible combina ions, we ha e elabo a ed a able ha summa ise hese esul s, and in appendix 1 g aphs o each case a e included: Si ua ion Dis u bance Cons an s Channel Del a 1 Vc*>0, Vd*>0 C1>0, C2>0 o e shoo mono one 2 Vc*>0, Vd*>0 C1<0, C2>0 o e shoo mono one 3 Vc*<0, Vd*>0 C1<0, C2>0 mono one mono one 4 Vc*<0, Vd*>0 C1<0, C2<0 mono one mon/bump 5 Vc*<0, Vd*<0 C1<0, C2<0 mono one mon/o e sh 6 Vc*<0, Vd*<0 C1>0, C2<0 o e shoo mono one 7 Vc*>0, Vd*<0 C1>0, C2<0 mono one mono one 8 Vc*>0, Vd*<0 C1>0, C2>0 mono one mon/o e sh Table 3 Ini ial esponse o each elemen A e his epo 4.4 Th ee elemen s in he idal sys em 4.4.1 Model equa ions In he case o h ee elemen s, hus channel, la and del a, an app oxima ion is also done linea ising he ASMITA equa ions. In o de o simpli y his p ocess, equa ions ha e been gene alised as i will be shown nex . Howe e , o keep in mind equilib ium ela ions be ween he elemen s, we will w i e down hose equa ions and we will p esen igu e 3-4 again: Chap e 4: Linea app oxima ion o ASMITA equa ions 50 ()() ( ) od d E dc d c c d sd d de d cc cc QcQcwAc c δδ −+ −− + = − (4.54) () () ( ) c c dc c d c sc c ce c cc cc QcSwAc c δδ −+ −+ −= − (4.55) () ( ) c c s e cc wAcc δ −= − (4.56) Q QQ Qw ww w () dc c d cc δ − S SS S Q QQ Qw ww wc cc cc cc c Q QQ Qw ww wc cc cd dd d Q QQ Qw ww w Q QQ Qw ww w () od d E cc δ − () c c cc δ − () e wA c c− () cc c ce wA c c− () dd d de wA c c− d y olume we olume la ou side wo ld del a channel Figu e 3-4 Exchange lines in case o i e discha ge A e his epo Fi s o all we will p esen he gene al exp ession o a sys em wi h n elemen s, and a e wa ds, apply i o ou case, he h ee elemen s sys em. ()()  () ,1 1 ,1 1 n nn nn nn nn snnnen cc cc a wAc c δδ ++−− −+ −+= − (4.57) () nn sn n ne n n dV wA c c d =− (4.58) n ne ne E n V cc V æö =ç÷ èø (4.59) 1,nN= ; and 2 n =± , “+” o we olumes, “-“ o d y olumes j is he elemen whe e he i e discha ge goes o N is he o al numbe o elemen s in he sys em Chap e 4: Linea app oxima ion o ASMITA equa ions 51     1 1,1 0, , ( ), 1 ( ) , n nn nnn nnn nnE anj acQS nj aQcc j nN aQcc c nN δ − −+ =< =− = =− +<< =−− = (4.60) whe e ,1nn δ +, in case nN=, co esponds o , N ou side wo ld No δδ = Using equa ions (4.58) and (4.59), we ob ain he gene al exp ession o n c: 1, whe e n ne n n nE n snn nn n VdV cc awA Vad æö =− = ç÷ èø (4.61) And, subs i u ing his equa ion o he i s one we ob ain a di e en ial equa ion whe e he e a e no inpu pa ame e s unknown: ()  1 1 ,1 ,1 ,1 ,1 11 11 11 ,1 ,1 ,1 ,1 11 nnn nn nn nn nn nnnn nnnnn ne ne ne n nn E nn nn E nn E nnn dV dV dV ad a a d ad VVV ccca VVV δδδ δ δδδδ + − ++− − +− +− +− −+−− +− éù +− − − + = êú ëû æö æö æö =−++− ç÷ ç÷ ç÷ èø èø èø (4.62) Le 's w i e he p e ious exp ession again in he h ee elemen s case, and o each elemen : 21 21 12 2 1 1 12 12 12 21121 1 ( la ) 0 ee EE n VV dV dV cc ad d a V V δδ δδ = éù æö æö +−−− + = êú ç÷ ç÷ èø èø ëû (4.63) () 32 12 23 3 23 3 2 2212211 23 23 21 322213 2 122 21 122 2 (channel) ee EE ee EE n dV V V dV dV cc ad d a a ad V V VV dVQ cQc S VVad δδ δδ δδδ δ = éù æö æö + −−− + − + + − êú ç÷ ç÷ èø èø ëû æö æö −+−= ç÷ ç÷ èø èø (4.64) Chap e 4: Linea app oxima ion o ASMITA equa ions 58 become meaning ul in he case o he channel. Also, we can see he olumes seem o be cons an o high discha ge alues, hence, i seems he e would be no maximum i e discha ge. The linea ised equa ions a e alid o an in e al a ound an equilib ium s a e whe e he e is no i e . The e o e, as la ge is he i e discha ge, as much u he we a e om his equilib ium olumes, and he i s o de equa ions ha e less alidi y. In o de o be e compa e las wo igu es, we p esen a g aph wi h bo h esul s a he same ime, only o he channel olume e olu ion: -0,5 0 0,5 1 1,5 2 2,5 3 3,5 0 100 200 300 400 500 600 700 Q (m3/s) ne channel olumes (m3) channel non lin channel lin Also, o high discha ges, he linea ised equa ions do no gi e enough p ecision since hey a e a i s o de app oxima ion o second o de di e en ial equa ions made a ound a ce ain poin . And, he app oxima ion is good wi hin an in e al no oo wide a ound his poin . Then, in case o oo high discha ges, his is no a good app oxima ion because olumes a e oo a om he ini ial equilib ium si ua ion, om he poin a ound which we app oxima e. In his case, linea isa ion should be made a ound ano he poin : he end olume, since i is he alue a ound which elemen s end o e ol e. Fu he mo e, a ela i e e o (ε) can be calcula ed as we a e able o ob ain he “exac ” end olumes, ha is wi hou a simpli ica ion, acco ding o he ollowing exp ession: ,, , end lin end ull end ull VV V ε − = (4.80) Figu e 4-21 Compa ing ull equa ions wi h linea ised equa ions esul s A e his epo Chap e 4: Linea app oxima ion o ASMITA equa ions 59 whe e Vend,lin co esponds o he end olume ob ained wi h he linea ised equa ions ( he app oxima e olume) and he e m Vend, ull co esponds o he end olume in case we use he ull equa ions. Nex , we will p esen a g aph whe e one can see how he ela i e e o inc ease, specially o he channel olume, when he discha ge also inc eases: 0 0,2 0,4 0,6 0,8 1 1,2 0 100 200 300 400 500 600 700 Q (m3/s) Rela i e e o la channel del a Figu e 4-22 Rela i e e o depending on he i e discha ge A e his epo We migh no ice ha Vi,eq=Vi,end when he e is no i e discha ge (Q=0). I becomes clea ha he discha ge is wha makes he sys em be mo ed om he o iginal equilib ium si ua ion. The e migh be a balance be ween he alue o he i e discha ge and he p ecision in sol ing he linea ised ASMITA equa ions. As much la ge is he i e discha ge, as much wo h i is o ake in o accoun his bounda y condi ion in he model o mula ion o be close o eali y, bu a he same ime, he solu ion has less exac i ude. A comp omise be ween bo h si ua ions migh exis . Howe e , exp essions o he end olumes wi h in case o ull equa ions a e easy o ob ain ( hey will be p esen ed in nex chap e ), no ma e how la ge is he i e discha ge. This becomes a use ul ool in ha ing an idea abou he e o made in he gene al use o he linea ised equa ions.