Geology 607: Advanced Physical Hydrology

Transcription

1 Gology 67: Avnc Physicl Hyrology Lcur Nos for Fll 7 Insrucor: Mski Hyshi offic: ES 78, phon: , -mil: hyshi@uclgry.c xooks: Jury, W.A. n Horon, R., 4. Soil physics, 6 h iion. John Wily & Sons. Brusr, W., 5. Hyrology: n inroucion. Cmrig Univrsiy Prss. Dingmn, S.L., 5. Physicl hyrology, 3 n iion. Wvln Prss. McCun, R.H., 5. Hyrologic nlysis n sign, 3 r iion. Prson Eucion. CHAPER : WAERSHED AS A LEAKY RESERVOIR. Bsic Concps Hyrogrph is im-sris plo of srm ischrg, Q m 3 s - Somims Q is normliz y h rsh r A m. R = Q /A R is runoff m s - or mor commonly in mm y - Srm hyrogrph ingrs complx hyrologicl procsss, in priculr: Hillslop procsss Chnnl procsss Rliv impornc of incrss ih h siz of rsh s Fig..5 in Dingmn.. Hyrogrph Sprion riionlly, physicl hyrology s vlop for o purposs: Pricion of floo mgniu n iming Dsign n oprion of r supply rsrvoirs Much of rly ffors r m ors h nlysis of high flo uring sorm n sno ml vns. Qusion: Givn n moun n iming of rin, ho much of i ill com ou o srms, n ho fs? o nsr his usion, on oul hv o spr h flo rsuling from sorm r inpu from h sflo.

2 A numr of grphicl mhos hv n us o spr sflo from sorm flo s McCun, p No h ll of hs mhos r rirry ihou rl physicl sis. Shoul rgr s convnin ficion. Consn slop mhos McCun, p.495 ssum h h s flo componn linrly incrss uring n vn. homork ssignmn # his n similr concps r ily ccp y nginring hyrologiss; hir vliiy s rrly usion unil h rly 97s. hn rvoluion hppn mor on his lr. oy i is ccp y mos scinific hyrologiss h h sorm flo componn conins lrg frcion of pr-vn r, hich is prsumly push ou y vn r..3 Sorm Runoff Volum On cn sim h runoff volum y ingring h sorm flo componn from o in h figur ov. Dno i Wff, ffciv r inpu P in Dingmn, p In gnrl, i is much smllr hn h ol r inpu y rin or sno ml = W Exmpl: W = prcipiion rsh r h rio Wff/W is ofn cll runoff rio. Why is Wff/W <<.? - Iniil srcion: cnopy inrcpion, prssion sorg, c. - N o ovrcom soil moisur fici. - h r conriuing runoff is smllr hn h gross ring r riv from igil lvion mol DEM. Mor on h conriuing r lr. In h folloing nlysis, ssum h Wff is knon. Ho?

3 3 hr r numr of mhos o sim Wff from prcipiion inpu s Dingmn, p. 54. Agin, hs r ll convnin ficion. W ill us h rionl mho Fig. -49 in Dingmn: = p CR : im vrying ffciv r inpu p: rinfll innsiy msur CR: mpiricl cofficin, CR = Wff / W = ol sorm runoff / ol prcipiion his mho is oprionlly vry simpl, u i ill proly ovrsim rly in h sorm, n unrsim lr in h sorm. Why?.4 Inpu-Oupu Rspons Funcion Approch In his pproch, rsh is consir lck ox h proucs srm hyrogrph in rspons o ffciv r inpu Dingmn, p Whil his os no giv us much insigh ino hyrologicl procsss, i os srv som usful purposs: I hlps us xmin h hyrologicl flux in uniiv mnnr. Shp of h rspons funcion rvls som insighs ino rsh chrcrisics. 3 Whn clir, i cn us for priciv purposs. 4 I provis us ih moivion for furhr suy. h rspons funcion is lso cll rnsfr funcion or krnl funcion in h lirur. Mny forms of rnsfr funcions hv n propos, u ill xmin h simpls of ll. Linr rsrvoir or nk mol s Dingmn, p.47. Bsic concp is h: h sysm is simpl nk ih n oul. h ouflo r is proporionl o h volum of r in h nk.

4 4 Mhmicl Inrlu M. Mss Blnc Concp n Diffrnil Euions In h simpl rsrvoir: inpu - oupu = r of sorg chng V ΔV = volum chng, Δ = im inrvl Whn k Δ n ΔV o infinisimlly smll vlus: V V lim hrfor,, V V From our sic ssumpion i.. is proporionl o V: V hr / is proporionliy consn. hs imnsion of im hy?, n is cll rspons im of h sysm. h mss lnc uion cn rin s: No h hr is only on inpnn vril,, in his sysm, so n orinry riviv is us on h righ hn si. Also, is no pnn on. Such n uion is cll linr orinry iffrnil uion ODE of firs orr. h uion oul non-linr, if or r pnn on. M. chnius for Solving Firs-Orr Orinry Diffrnil Euions Suppos yx, funcion of x, hich sisfis: y N x, y M x, y M- x Whr Mx,y n Nx,y r ny funcion of x n y..g. Mx,y = 3x + xy Nx,y = x + y

5 5 All firs orr orinry iffrnil uions, linr or non-linr, cn rin in his form. Suppos spcil cs, hr Ny is only pnn on y, n Mx only pnn on x: y i.. N y M x x In his cs, cn spr h vrils: Nyy + Mxx = M- hn ingr oh sis: N y y M x x C hr C is n ingrion consn. No h h symol M x x is cll infini ingrl or ni-riviv; i is h invrs oprion of iffrniion. By finiion, x M x x M x h ingrion consn ns o rmin using n iniil coniion, such s: y = y hn x = x Alrnivly, cn ingr M- mploying h iniil coniion y y N x x M h Grk symols r cll ummy vrils of ingrion. Sinc h uppr limis of ingrls r h vrils of inrs x n y, i is ssnil h ummy vrils us for ingrion. x No h, gin y finiion, M M x x y n N N y x x y Exmpl: y Ky rx ih h coniion y = y x = M-3 x hr K n r r consn.

6 6 If inify K s hyrulic conuciviy n r s r-l rchrg flux, M-3 is h Dupui-Forchhimr uion scriing h sy-s grounr flo in r-l uifr. is h isnc from h ring ivi. o solv M-3, firs spr y n x: Kyy + rxx = Ingr h ov from, y o x,y: K y r x y K y y y y r K x r x For gnrl css of Mx,y n Nx,y, n o us mor vnc mhos, hich r yon h scop of his cours. M.3 Firs-Orr Linr Orinry Diffrnil Euion h mos gnrl form of firs-orr linr orinry iffrnil uion is: y P x y Q x M-4 x I cn shon h homork ssignmn, h gnrl soluion o M-4 is givn y: Px Px Px y Q x C M-5 Px hr is shor-hn noion for xp[ P x x] n C is n ingrion consn o rmin using h iniil coniion. Px rprsns h propry of h sysm n Qx rprsns h forcing, oh of hich r inpnn of y, hich is h sysm s s vril. For priculr prolm ih prscri Px n Qx, M-5 cn vlu nlyiclly for simpl funcions, or numriclly for complx or iscr funcions. No: h ckgroun informion on M.3 cn foun in mos inroucory xooks on ppli iffrnil uions, for xmpl Spigl, M.R. 98. Appli iffrnil uions, 3r iion, Prnic-Hll, pp A copy of h rlvn pgs is vill on h cours si.

7 7.5 Linr Rsrvoir Rspons Funcion As iscuss in pg 4, h ODE scriing simpl linr rsrvoir is givn y s Dingmn, p. 47: - hr is im-vrying inpu funcion [L 3 - ] is oupu flux from h rsrvoir [L 3 - ] is sysm rspons im [] No h h imnsion of h uion [L 3 - ] is h sm in oh h lf hn si LHS n righ hn si RHS. imnsionl homogniy his is n imporn ruirmn for corrc uions rprsning physicl sysm. o solv -, r-ri i s - n impos h iniil coniion: = = -3 Euion - is firs orr linr orinry iffrnil uion, for hich h soluion cn foun using M-5. W inify: P Q P P Q Susiuing hs ino M-5 yils: C o rmin h ingrion consn C, ri h ingrl s fini ingrl. C No h h ummy vril mus us s h ingring vril, cus pprs s h uppr oun of h ingrl.

8 8 Sing = in h ov givs: = C hrfor, C = from h iniil coniion -3. Noing h in h ling rm is consn ih rspc o, cn mov i ino h ingrl: hrfor, -4 Euion -4 sisfis oh - n -3 W hv foun h soluion! h soluion cn rin s: u hr u his yp of ingrl is cll h convoluion ingrl. is h ummy vril of ingrion. is commonly rfrr o s inpu or forcing funcion. u is rfrr o s rspons or rnsfr or krnl funcion. In mnnr of spking, rplc h lck ox in pg 3 y u. W sill o no kno h is in h ox, u kno ho i mhmiclly rspons o n xrnl forcing. For spcil cs hr = consn, cn ri E. -4 s: -5 For h nlysis of sorm flo lon, usully hv spr h sflo i.. iniil flo for h ons of h sorm = hrfor, h sorm flo porion of h hyrogrph is givn y: -6

9 9 in hich, kps incrsing s long s h inpu = is hr. For sorm lsing for urion pk, h mximum flo is: pk pk shor-hn noion pk [ xp{ }] long-hn noion h rcssion pr of h hyrogrph > pk is givn y sing = n = pk in E. -5: pk -7 pk No h h rcssion ill plo s srigh lin on smi-log grph cus: pk ln ln pk pk log log pk.33 No h: ln.33 n lnx = ln logx In h xponnil rcssion hyrogrph, rmins posiiv for, u i coms ngligily smll for lrg. Ho lrg is lrg nough o cll ngligil? I is rirry, u Dingmn p. 47 suggss / pk =. s cuoff; i.. pk. pk = ln. or pk h urion of h sorm il is proporionl o h sysm rspons im. R Dingmn pg for ohr propris of Es. -6 n -7 s ll s hir finiions.

10 .6 Uni Hyrogrph Suppos sorm vn ih consn inpu r n urion. For rsh chrcriz y h linr rsrvoir rspons uions -6 n -7, cn gnr sorm hyrogrph rsuling from his vn. From uions -6 n -7, i is clr h h shp i.. mporl isriuion of h hyrogrph is inpnn of. h mgniu of is proporionl o. If kno h sysm rspons for uni moun of inpu, hn cn clcul h sysm rspons for ohr mouns of rin. riionlly, h uni moun of inpu is fin y on inch = 5.4 mm of inpu isriu uniformly ovr. For xmpl, h figur o h righ shos hypohicl sorm in hich 5.4 mm of inpu is uniformly isriu ovr -hr prio. Hyrogrphs for non-uniform r inpu cn uil from ing h conriuion rsuling from ch hr of r inpu s h figur lo. lso s McCun pp o o his compuion on compur, n n i u i ni hr i is h r inpu im i un i is h uni hyrogrph vlu im n i lrgr h n i, smllr h vlu -9 Dils of h numricl lgorihm r scri in McCun pp In his clss, ill us MALAB funcion o vlu E. -9. homork ssignmn Rcll from pg, hn =, h ouflo from linr rsrvoir ih inpu is:

11 u - hr u is h rspons funcion. I cn shon h Es. -9 n - r uivln. E. -9 is h iscr vrsion of h coninuous convoluion ingrl -. h concp of uni hyrogrph is hlpful for unrsning h physicl mning of convoluion. Imporn unrlying ssumpions for Es. -9 n - r: - h rspons funcion or uni hyrogrph is inpnn of h siz or urion of sorm i.. h sysm is linr - h rsh chrcrisics o no chng ih im. i.. h sysm is sionry hs ssumpions r rrly sisfi in h rl rsh. Why? s Dingmn pp hrfor, h us of Es. -9 n - ruirs cuion. Es. -6 n -7 r jus on xmpl of simpl rspons funcion or uni hyrogrph. s McCun pp for mor complx forms of uni hyrogrphs..7 Rspons o Prioic Oscillion of W s spcil cs of E. -4 ih consn r inpu in pg 8. Anohr spcil cs is prioic oscillion of..g. iurnl sno/glcir ml inpu iurnl ngiv inpu y vpornspirion Assum h = + cos hr: [L 3 - ] n [L 3 - ] r consns [ - ] is h ngulr fruncy of oscillion; p [] is h prio of oscillion,.g. 4 hr p Susiuing his ino E. -4 n sing =,

12 cos cos Noing h sin cos sin cos cos hrfor, sin cos sin cos Noing h is ngligil for >>, sin cos - I cn shon homork ssignmn h E. - is uivln o cos p hr n p - Euion - inics h: hs smllr mpliu hn mping fcor = / p hs phs lg of, compr o pk of is ly y p / hs cn us o xrc h informion rgring sysm rspons im from h osrvion of prioic oscillion in.

13 mor on his lr, hn iscuss Fourir rnsform n hrmonic nlysis. 3