Mathematical Function of Physical Variable. and Material Transport Deposition Map in the. River Estuary

Transcription

1 Advaced Studes Theoretcal Physcs Vol. 8, 04, o. 3, HIKARI Ltd, Mathematcal Fucto of Physcal Varable ad Materal Trasport Deposto Map the Rver Estuary Max Tedea Departmet of Physcs Faculty of Mathematcs ad Natural Sceces State Uversty of Maado, Idoesa Copyrght 04 Max Tedea. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Abstract The assessmet of rver estuary physcal eroso ca be based o posto ad tme dstrbuto of velocty ad sedmet trasport. The physcal varables are partally or tegratedly used to assess the flow characterstcs. Aalyss of the physcal varables ad ther chages alog the rver estuary ca show the process of sedmet deposto the estuary. Data dstrbuto ad flow physcal varable profle wll geerate mathematcal fucto of chages alog the estuary. Spatally, ths mathematcal fucto wll geerate ad determe the posto of the materal trasport deposto (map of sedmet deposto alog the estuary of the rver). The materal trasport deposto map (based o the valueof (c0), presets the cotours of c0 durg the low tde (preseted the form of curves/ full le), whle the cotours of c0 durg the hgh tde (s preseted the form of dashed les). The cocetrato cotour of materal trasport at hgh tde s at the posto of 80 meters of c0 equals.35 grams per ltter, at low tde c0 equals.36 grams per ltter, ad at the maxmum hgh tde, almost all sedmet s the form of fe sad sedmet deposted at the posto of 0 meter up to the posto of 80 meters, the c0 value s equal to.6 grams per ltter. It ca be see the cotour map the shftg posto or cotour les at low tde ad hgh tde, ad whe the tde s shfted farther to the upstream of shorele. Cotour les shft to the upstream

2 004 Max Tedea drecto of shorele at hgh tde shows the posto shft of deposto of materal trasport to the upstream drecto. I ths codto, the cocetrato of sedmet the form of fe sad at the maxmum hgh tde, almost all of t s deposted at the locato ext to shorele upstream. Keywords: mathematcal fucto, physcal varable, sedmet deposto. Itroducto The process of sltg ad eroso of the rver estuary s geerally the result of deposto or eroso ad trasport of sedmets process the Earth's surface (Dbyosaputro, 997, Masselk et al, 009). The materal trasport whch determes the formato of deposto results clude bed load, suspeded load, ad wash load. The materal deposto rate s very depedet o the velocty of the stream that trasports t. The stream velocty s also a determg factor of rverbed eroso rate domated by other tha the type ad dmesos of the rverbed materals such as fe sad accordg ts compacto level. The measuremet requremets of flow physcal varables lke sedmet trasport ad velocty, the steady flow codtos, o-turbulet or rrotatoal, ad homogeeous, are very dffcult to carry out so that the measuremet must be coducted at may pots of observato (Ve Te Chow, 984). The methods of measuremet ad determato of measuremet pot (accordg to the depth ad wdth of the rver estuary) should be coducted such way that the flow requremets are met. The physcal assessmet of deposto of materal sedmet trasport alog the rver estuary ca be based o the dstrbuto of the posto ad tme of both physcal parameters.whle the physcal varable of stream velocty s partally or tegrated used to assess the characterstcs of stream ad sedmet trasport deposto process alog the estuary (Bartk et al, 99). Aalyss of these varables ad ther chages alog the rver estuary ca show sedmet deposto process (materal trasport) the estuary area. The avalablty of data ad the physcal varable profles ca geerate mathematcal fucto of velocty chages ad fucto of sedmet trasport cocetrato alog the estuare flow, ad the mathematcal fucto of materal trasport cocetrato chages alog the estuary, are formulated equatos of pot coordates to map the sedmet cocetrato alog the stream wth the posto of pots that have the same cotour. The cotour map of materal trasport cocetrato that determes the deposto ad trasport of sedmet at the estuary, the becomes mportat, because the rver estuary s a area of eroso of the rver bed ad deposto area of sedmet materal, suspected to be able to result the alluval formato of both buldg estuare bed surface ad lower the estuare bed surface.

3 Mathematcal fucto of physcal varable 005 Theoretcal Bass Of most rvers, estuary s the depostoof materal or trasport sedmet carred from upstream areas (Wdyato 986, Dbyosaputro, 997, Mulyato H.R., 00;). Stream codtos ad the tdal actvtesat rver estuary area produce alluval formatos such as alluval plas, deltas, sadbars, lagoos etc. The potetal of the formato above ca be physcally examed through materal trasport mechasm, ad coversely, the observato of formato chages over tme ca be assessed ts process ad the flow physcal characterstcs varables the estuary ca be determed. Materal trasport as a atural process s closely related to trasportato, deposto, ad compacto, especally lad areas or places where water flows. I prcple, materal trasport (sedmet) relates to the sedmet dscharge trasported by the stream ad velocty of stream trasportg the sedmet. Progressvely accumulated, the trasported materal trasported o the rver bed ad flood pla or body of water where the rver stopped s depedet o sedmet dscharge how fast the sedmet carred by the stream. Therefore, sedmetato occurs as a result of gradet of rvers or small surface slope, ad small rvers flow dscharge, tll the trasportg o loger able to trasport sedmet (Dbyosaputro, 979). I geeral, materal trasport deposto o rverbed ad alog the rver bas occurs at low velocty or flow dscharge, or crcumstaces whch the water level s low. Sedmet remoblzato occurs durg hgh or creasg dscharge perods or o the hgh water level (De Boer, 99). The amout of materal trasport stream s also a fucto of sedmet supply ad stream eergy. Whe the magtude of the stream eergy exceeds the magtude of sedmet supply, there wll be degradato of rver. O the other had, whe the sedmet supply s greater tha the stream eergy, there wll be aggradato of rver (Asdak, 007). 3 Method Dckso ad Bolto (99), state that bed materal load ad wash load are measured ad calculated separately. Wash load s estmated to be evely dstrbuted throughout the cross secto of the rver, ad ca be calculated drectly as the result of multplyg sgle measured cocetrato wth the average dscharge at the samplg tme. Bed load (bed materal trasport) s determed more by the hydrologcal parameters so that data collecto ca be smplfed by developg a "sedmet ratg curve" that coects the average sedmet cocetrato wth dscharge, for measuremets a short perod. Accordg Zha ad Gagya (99), although may methods of calculato have bee developed for the determato of materal trasport, but there s o a sgle calculato that ca be used uversally for all szes of sed-

4 006 Max Tedea met, bed cofgurato, ad flow regmes. The ma dffculty les the calculato method of determg partcle sze dstrbuto movg o the rverbed. Bartk et al. (99) states that the materal trasport occurs whe the flow dscharge exceeds the crtcal value. It s ot easy to determe the flow dscharge o whch materal trasport wll beg to move. The research results show hgh coformty betwee the results of expermets ad materal trasport equato formulated by Meyer-Peter & Muller modfed by Wag. I 935, through laboratory research, Mavs developed a formula to determe the velocty of partcle (materal trasport) relatvely more flexble for varous partcle dameter from 0.35 to 5.7 mm ad a specfc gravty from.83 (mm.sec - ) to.64 (mm.sec - ): vt = 0,5 d 4/9 (G ) / () o f whch vt s partcle velocty (m sec - ), d s partcle dameter (mm), ad G s specfc gravty (m sec - ) (Schwab et al, 98). If the flow velocty exceeds vt value (for the value ofd ad G s accordace wth bed materal of rver surface), the partcles o the surface of rverbed wll be trasported by the flow to more dowstream area, coversely, f the flow velocty s lower tha vt value, the the surface of rverbed wll ot be eroded ad partcles o the surface of rverbed s ot trasported.. The velocty ecessary to precptate bed charges partcles (materal trasport) deposto velocty wll follow the equato: v0 = 0.55 m y () of whchv0 s the crtcal velocty, m s crtcal value rato (cvr), whch depeds o the type of deposto materal, ad y s water depth (Garg, 979). 4 Results ad Dscusso Materal trasport s a compoet of sedmet trasport that moves o the layer ear the surface of rverbed (bed stream). Thus, for the purposes of aalyss, flow velocty measuremet s carred out o the bed of rver estuary, (v0). For the ma flow path at a partcular measuremet posto, there are two measuremet segmets (dfferet secodary flow paths), the the velocty data at that posto (v0) s defed as the mea velocty data o both segmets. The modellg of mathematcal fucto of flow velocty of v0 materal trasport deposto determat at low tde ad hgh tde, results a expoetal fucto (egatve), wth the form of equato: v k k e kx 0... (3)

5 Mathematcal fucto of physcal varable 007 of whch v s the velocty the ut of cm.det -, k0, k, ad k are costatscoeffcets of fucto. x s the estuary elogated dstace. To solve ths equato, we eed the formulato of o-lear regresso followg theorem: Theorem: Estmato of Flow Velocty If gve the data pars followg predctor ad respose varables ( x, v ), f k0 s gve, the o-lear regresso model for velocty data s: ˆ ˆ... (4) kx v k k e 0 The estmato of parameters s gve: ˆ ˆ k exp L( v k0) k x x L( v k0) x ( L( v k0)) ˆ k x x Proof: Re-modelg Equato (4) to the lear regress model as equato (5) L( v k ) Lkˆ kˆ x... (5) 0 If y L( v k0), â ˆ Lk, ad â ˆ kthe the equato (5) equals to equato (6) as follow y aˆ aˆ x... (6) The parameter estmator â ad â usg the Ordary Least Square (OLS): ˆ ˆ y a ax... (7) M M ( ) Let s ( y aˆ aˆ x ), the the m of s acheve by partally dervatg s

6 008 Max Tedea agas â ad â equated to zero, as follow: ds daˆ ( y aˆ aˆ x )( ) 0 ( y aˆ aˆ x ) 0 y aˆ aˆ x...(8) ds daˆ ( y aˆ aˆ x )( x ) 0 ( y aˆ aˆ x )( x ) 0 x ˆ ˆ y a x a x 0...(9) Substtuto equato (8) to (9), the, the parameters of estmato are gve: aˆ x y x y x x ˆ aˆ y a x The codto k ˆ ˆ a, kˆ ˆ exp( a), ad y L( v k0), the the parameter estmato of ˆk ad ˆk are: kˆ ( ) ( ( )) x L v k0 x L v k0 ˆ a x x ˆ ˆ k ˆ exp( a ) exp L( v k0) k x

7 volocty (cm/sec) Mathematcal fucto of physcal varable 009 Ratg curve v0 aalyss at hgh tde shows that the flow velocty at the bed layer becomes zero at the posto more to the shorele upstream. Physcally, because the desty of sea water s hgher tha the desty of rver water, the boudary meetg profle of both water mass has the decreased gradet to the shorele upstream drecto. I these codtos, lower water mass layer wll experece decelerato at a posto more to the shorele upstream ratg curve vo low tde ratg curve vo hgh tde Dstace (let PDAM (m)) Fgure -. Ratg Curve of v 0 Velocty (Path - ) The materal trasport cocetrato s oly measured the layer ear the surface of the rverbed. Flow cocetrato s measured at low tde ad hgh tde, the same posto wth the posto of the velocty measuremet. As well as the velocty, materal trasport data s aalyzed per path of flow. The data modellg of each flow path for low tde ad hgh tde result mathematcal models fucto of materal trasport deposto, wth egatve expoetal equato form: c0 = k0 + k e - kx... (0) of whch x s the dstace of posto measuremets from the referece. The data modellg aalyss of materal trasport cocetrato dcates ratgcurve result at hgh tde greater tha at low tde ad t ca be cocluded that the cocetrato fucto gradet at hgh tde s steeper tha the fucto gradet at low tde.

8 bed load (gr.ltr-) 00 Max Tedea,60,40,0,00 0,80 0,60 0,40 0,0 0,00-0,0 ratg curve co low tde ratg curve co hgh tde Dstace ( let PDAM (m)) Fgure-. Ratg curve of materal trasport cocetrato at low tde ad hgh tde (path-) Ratg curve at low tde, the materal trasport cocetrato s decreased wth slghtly slopg gradet from upstream toward the shorele. I ths codto, most sedmet s deposted a posto more to the shorele from the posto of 655 meters, eve partally of t passed o to the sea. At hgh tde, the gradet ratg curve of materal trasport cocetrato becomes steep ad o the posto approachg 600 meter of the measuremet posto of oe, the flow cocetrato s almost zero. If the sea level s creased, the gradet s gettg creased ad the boudary of deposto posto wll be shfted towards the upstream, coversely, f the sea level goes dow, the gradet wll be decreased ad the posto of deposto wll be shfted towards the shorele. Based o the mathematcal fucto form, the materal trasport deposto form s expoetal fucto (equato 4), the there s derved equatos to determe the cotour map of materal trasport deposto alog the estuary, pot posto that has the same cotour of materal trasport cocetrato s obtaed by the equato: x = ( l k - l(c0 k0))..(5) k of whch x s the pot posto wth the same value of c0cocetrato, k0, k, ad kare costats ad coeffcets of cocetrato fucto. The cotour map of sedmet trasport cocetrato (based o c0 value), presetg cotour of c0 at low tde s preseted the form of curves/ full le, whle the cotour of c0 at hgh tde s preseted the form of dashed les. The cotour map shows the shft posto or cotour les at low tde ad hgh tde. Examples for the value of c0 =.6 at low tde, the posto s located at 80 meters, ad at hgh tde t s shfted further to the upstream, that s at posto 90 meters from the referece. The cotour le shft towards the upstream at hgh tde shows the shft of sedmet deposto posto to the upstream toward the shorele. The cotour of

9 Mathematcal fucto of physcal varable 0 sedmet trasport cocetrato at hgh tde at the posto of 80 meters c0=.35 grams per ltter at low tde c0 =.36 grams per ltter, ad at the maxmum hgh tde, almost all sedmet s the form of fe sad deposted at posto of 0 meter to 80 meters of co value equal to.6 grams per ltter kotur c o low tde kotur c o hgh tde stadardt posto 5 Cocluso Fgure 3. Sedmet trasport map alog the rver estuary. At low tde, the materal trasport cocetrato s decreased wth slghtly slopg gradet from upstream toward the shorele. At low tde, most of the sedmet trasport s deposted at a posto that s located more towards the shorele, eve partally of t s passed o to the sea. At hgh tde, gradet ratg curve of sedmet trasport cocetrato becomes steep ad at the posto to the upstream, the flow cocetrato s almost zero. At hgh tde, all bed loads are deposted at the posto more to the upstream of the posto of 600 meters. Physcally, cocetrato ratg curve gradet at hgh tde wll be chaged, depedg o sea levels. If the sea level s creased, the gradet s gettg creased ad the lmts of deposto posto wll shft towards the upstream.

10 0 Max Tedea The sedmet trasport cotours map (based o the value of c0), presetg the cotours of c0 at low tde s preseted the form of curves/ full le, whereas the cotours of c0 at hgh tde s preseted the form of dashed les. I the cotour map, t ca be see the posto shftg or cotour les at low tde ad hgh tde, ad at hgh tde t shfts farther to the upstream. The shft of cotour le towards the upstream at hgh tde shows the shft the materal trasport deposto posto to the upstream. The materal trasport cocetrato cotour at hgh tde s the posto more to the upstream of shorele (c0 = 0.05 grams per ltter) showg that at the maxmum hgh tde, almost all sedmet s the form of fe sad (bed materal trasport) deposted the locato ext to shorele upstream. Refereces [] Asdak, C Hdrology ad Watershed Maagemet. Yogyakarta: Gadjah Mada Uversty Press ( Idoesa). [] Bartk W., M. Madeysk ad A. Mchalk. Suspeded Load ad Bed Load Trasport Mouta Sreams Determated Usg Dfferet Method : :Procedg of the It Symposum o Eros ad Sedmet Trasport Motorg Programmes Rver Bas Oslo, Norway, 4 8 August 99, page 3-9. [3] De Boer D.H. (99). Suspeded Sedmet Dyamcs of Rvere Lake of the Lawrece Rver, Caada. Procedg of the It Symposum o Eros ad Sedmet Trasport Motorg Programmes Rver Bas Oslo, Norway, 4 8 August 99, page (473-48). [4] Dbyosaputro, S Study of Rver Water Sedmet Yeld Wlukulo Hulu Watershed above AWLR Karagsambug Kebume. Udergraduate Thess. Yogyakarta; Geography Faculty of UGM ( Idoesa). [5] Dckso A ad Bolto P. 99: A Program of Motorg Sedmet Tras [port North Cetral Luzo, Phlpa. Procedg of the It Symposum o Eros ad Sedmet Trasport Motorg Programmes Rver Bas Oslo, Norway, 4 8 August 99, page [6] G. Masselk, L. Cotre, J. Wllams, R. Gehrels ad W. Blake, Tdedrve due mgrato ad sedmet trasport o a tertdal shoal a shallow estuary Devo, UK, Mare Geology, Vol. 6, pp. 8-95, 009. [7] Garg S.K., 979: Water Resources ad Hydrology (thrd ed.), Khaa Pub, - B, Nath Marlet, Na Sarak, Delh Ida.

11 Mathematcal fucto of physcal varable 03 [8] Mulyato H.R., 00., Prcples of Estuary ad Coastal Cotrol Egeerg, Graha Ilmu, Yogyakarta ( Idoesa). [9] Wdyato, 986, Geomorfolog Daerah Glaga-Bogowoto, UGM Yogyakarta. [0] Zha X., ad Gagya Z., 99. Measurg Techque of Bed Load the Yagtze Rver. Procedg of the It Symposum o Eros ad Sedmet Trasport Motorg Programmes Rver Bas Oslo, Norway, 4 8 August 99, page Receved: August 6, 04