A Kolmogoroff-type Scaling for the Fine Structure of Drainage Basins

Transcription

1 Kolmogooff-type Scaling fo the Fine Stcte of Dainage Basins Gay Pake, Uniesity of Minnesota Pete K. Haff and. Bad May, Dke Uniesity

2 Kolmogooff scaling in tblent flows: Enegy Cascade Tblent enegy is podced in eddies scaling with the size of the "box" (e.g. ie depth). The nonlinea tems in the eqations of momentm balance gind lage eddies into ee smalle eddies. The ginding contines ntil the eddies ae so small that the tblent enegy can be effectiely dissipated into heat. Thee ae no smalle eddies. podction of tblent enegy at lage scales nonlinea tansfe of enegy fom lage to small scales dissipation of tblent enegy to heat at small scales

3 Balance of enegy in a steady tblent flow (mean kinetic enegy of tblence) t podction ate dissipation ate + spatial tansfe ate Let chaacteistic tblent elocity (L/T) L size of the containe (e.g. ie depth) (L) ν iscosity of flow (L /T) P podction ate/mass of tblent enegy (L /T 3 ) ε dissipation ate/mass of tblent enegy (L /T 3 ) k Kolmogooff elocity scale (L/T) η k Kolmogooff length scale P 3 i i i k i j ~ ε ν ~ ν x j L x j x j ηk

4 Make Kolmogooff scales fom the iscosity ν (L /T) and the dissipation ate ε (L /T 3 ) 1/ 4 3 ν 1/ 4 kηk ηk k ( νε) ε ν Recalling that 1 P ε ε ~ 3 it is fond that η k ν ~ L L 3 / 4 That is, the highe the Reynolds nmbe of the flow the fine is the tblence L

5 Fom Tennekes and Lmley

6 Does the idea of a Kolmogooff scale hae any application to dainage basins?

7 How dense does a dainage netwok hae to be in ode to "coe" the catchment?

8 Well, how dense?

9 Conside a (statistically) steady state landscape ndegoing plift at constant ate (L/T). The channels ae ndegoing incision at ate I (L/T). The hillslopes contain a well-deeloped egolith which moes downslope with a kinematic diffsiity k (L /T) HYPOTHESIS 1: Channel incision ceates eleation flctations. HYPOTHESIS : Hillslope diffsion obliteates eleation flctations. HYPOTHESIS 3: The dainage netwok mst be jst sfficiently fine so that ate of ceation of eleation flctations balances ate of obliteation.

10 Channel incision in the nea-absence of hillslope diffsion: the SLOT CNYON: ceates amplitde flctation

11 Hillslope diffsion with weak channel incision: MELTED ICE-CREM MORPHOLOGY: destoys amplitde flctation Image cotesy Bill Dietich

12 Some scales L length of headwate tibtay L aea of headwate catchment: ~o(1) S h chaacteistic slope of hillslope in headwate catchment idge headwate tibtay

13 Some paametes η bed eleation t time (x 1, x ) spatial coodinate ζ incision ate (q 1, q ) ecto of olme hillslope tanspot/width k hillslope diffsiity q k η Exne eqation of sediment balance (neglecting poosity) idge η t ς + k η + headwate tibtay

14 Now fo simplicity we assme that ll the hillslope tanspot occs on the hillslope and ll the incision occs in the channel q η d t Integate Exne on hillslope qd k steady state η n ~ n ks h Hee n is nomal to ed peimete of path integal: gain de to plift idge headwate tibtay q n ds loss de to tanspot fom hillslope to channel

15 q Contine integation on hillslope: path integal is aond ed line η k ~ idge n n ks h q n ds ~ L q n ~ L ks h headwate tibtay The following scale elation eslts: L h L ks whee h is an o(1) paamete That is, the ate of hillslope dendation mst jst balance with ate of plift h

16 Incision ate: we assme ς I in channel 0 on hillslope Ths within the channel of the headwate tibtay η t steady state I + o ths Channel incision mst jst keep pace with plift I

17 Mean and flctating bed eleation η η η η η

18 Video clip Hasbagen and Paola

19 Eqation of eoltion of amplitde of eleation flctation Decompose into mean and flctating pats: η η + η ς ς + ς whee the oeba denotes an aeage oe an appopiate spatial window and the pime denotes a flctation abot that aeage. Local Exne: η t ς + k η +

20 Local Exne: η t ς + k η + Spatially aeaged Exne: Mltiply by η and edce: Mltiply local Exne by η, aeage and edce: B t t 1 η ςη + k 1 η + k + k + η t ςη + kη 1 ς + k η + η + ( η η) k( η) ( η) + η η ςη ς η ( η η) k( η) ( η) ( η η ) ( η ) ( k η ) η + η

21 t 1 Sbtact B fom to get eqation of eoltion of amplitde of eleation flctation: η ς η + k ( η η ) k( η ) ( η ) I II III IV I. Time ate of change of amplitde of eleation flctation II. Rate of ceation of amplitde flctation de to incision III. Rate of tanspot of amplitde flctation de to diffsion IV. Rate of destction of amplitde flctation by hillslope diffsion

22 Steady state: appoximate balance between ceation and destction of amplitde flctation ς η k ( η )( η ) Now if most of the destction is biased towad the finest scale of the dainage basin, i.e. the headwate catchment, then k d ks h ( ) ( ) η η whee d is an o(1) constant

23 ppoximate fom fo the ate of ceation of amplitde flctation by incision idge η p η c (t+ t) η c (t) I t headwate tibtay In the headwate catchment, appoximate the channel as incising into a plain with constant eleation η p. Channel eleation η c deceases with speed I. Ths: η η p + η c + BL BL whee B denotes channel width

24 Now 1 ( η ) 1 ( ηp η) 1 ( ηc η) on plain in channel Since η& c I it follows that η ς ( η η t c 1 ) ( η ) BL incision t 1 ( η ) + ( η ) plain t 1 + BL channel BL

25 Now scaling η ηc c H whee H is an "effectie" flow depth and c is an o(1) constant. Ths the ate of ceation of flctation amplitde by incision is expessed as η ς c H BL Balancing this against the ate of destction of flctation amplitde, c H BL d ks h

26 Scale elations fo headwate catchment: L Geometic scaling: h h ks L L h d c ks BL H Rate of hillslope dendation mst balance with ate of plift: Rates of ceation and destction of eleation flctation amplitde mst balance: 3 1/ 1/ e 1 1/ e k L 3 / 1/ e h k S whee e BH effectie channel aea and FROM THESE WE OBTIN 3 1/ h d c 3 1/ h d c

27 Let T the total aea of the dainage basin and L T ( T ) 1/ denote a length scale fo the total basin. It then follows that L L T 1 4 c d h 1/ 3 k 1/ e 1/ 3 L 1/ e T Now sppose that e is set. Then the atio L /L T becomes smalle (the dainage basin becomes moe inticate) as a) the size of the basin L inceases, b) the ate of plift inceases and c) the hillslope diffsiity k deceases. In addition, S h becomes lage (headwate hillslopes become steepe) as inceases and k deceases.

28 Sample calclation (nmbes fo k, cotesy Bill Dietich, Oegon Pacific Coast Range) "Kolmogooff" Scaling fo Dainage Basins c 5 channel incision paamete d 0.5 amplitde dissipation paamete B 5 m h hillslope diffsion paamete H m 0.5 geometic paamete fo headwate basin 0.1 cm/y k 300 cm^/y e / 3 L Re k m k L 7.68 m S h θ deg 1/ e 1 1/ 3 1 c h 4 d 1/ e S h k 1/ e c d / 3 h 1/ 3

29 Sample calclation (nmbes fo k, cotesy Bill Dietich, Oegon Pacific Coast Range) "Kolmogooff" Scaling fo Dainage Basins c 5 channel incision paamete d 0.5 amplitde dissipation paamete B 5 m h hillslope diffsion paamete H m 0.5 geometic paamete fo headwate basin 1cm/y k 300 cm^/y e / 3 L Re k m k L m S h θ deg 1/ e 1 1/ 3 1 c h 4 d 1/ e S h k 1/ e c d / 3 h 1/ 3

30 THE END!! O maybe not! Bt wait! It's not oe yet! Can we explain how sbmaine dainage basins fom? Tinity Canyon and associated dainage netwok, Eel Magin, Nothen Califonia

31 nd what abot the fine scale of tidal dainage netwoks? Banstable Salt Mash, Cape Cod, Massachsetts Image cotesy Tao Sn, Segio Fagheazzi and Daid Fbish

32 Sample calclation Conside two ies: a "pototype" with mean elocity U 4 m/s and depth H m, and a "model" with mean elocity U 1 m/s and depth H 0.15 m. One is a pefect Fode model of the othe. Note L ~ H, ~U/10, ν 1x10-6 m /s. Pototype η k ~ mm Eddies ange fom ~ m to mm Model η k ~ 0.11 mm Eddies ange fom ~ 0.15 m to 0.11 mm Scale model p to pototype: Eddies ange fom ~ m to 1.76 mm