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The for ma tion of sin gle-chan nel and mul ti ple-chan nel rivers on large slopes

Katarzyna MISIURA1, *, Leszek CZECHOWSKI1, Piotr WITEK1 and Anastasiia BENDIUKOVA1

1 Uni ver sity of War saw, De part ment of Phys ics, In sti tute of Geo phys ics, Pasteura 7, 02-093 Warszawa, Po land

Misiura, K., Czechowski, L., Witek, P., Bendiukova, A., 2016. The for ma tion of sin gle-chan nel and mul ti ple-chan nel rivers on large slopes. Geo log i cal Quar terly, 60 (4): 981–989, doi: 10.7306/gq.1330

We in ves ti gated the for ma tion of dif fer ent types of rivers de pend ing on slope, to tal dis charge, and grain size. Cal cu la tions were per formed us ing nu mer i cal pack age CCHE2D, de vel oped by the Na tional Cen ter for Com pu ta tional Hydroscience and En gi neer ing. The model is based on the Navier-Stokes equa tions for depth-in te grated two-di men sional tur bu lent flow and the three-di men sional con vec tion-dif fu sion equa tion of sed i ment trans port. For each model we use the same river ge om e try, sus pended load con cen tra tion, and bedload trans port rate. We dis tin guish three types of rivers us ing two meth ods for clas si - fi ca tion (sin gle-chan nel, mul ti ple-chan nels and tran si tional). We found that the trend line for tran si tional rivers is an in creas - ing func tion of Q in space (Q, d) and that for large S the num ber of mul ti chan nel rivers de creases.

Key words: river clas si fi ca tion, mul ti ple-chan nel rivers, sin gle-chan nel rivers, sed i men ta tion.

INTRODUCTION

River sys tems are im por tant fac tors that shape the sur face of the Earth. The anal y sis of river sys tems are based on dif fer - ent cri te ria. Gen er ally, four pure types of river are con sid ered:

sin gle-chan nel – straight and me an der ing; mul ti ple-chan nel – braided and anastomosing (e.g., Schumm, 1981; Nichols, 1999). In ad di tion to these “pure” types there ex ist also “in ter - me di ate” (tran si tional) rivers that have the prop er ties of a few pure types (e.g., wan der ing rivers – Schumm, 1981).

Lane (1957) and Leopold and Wolman (1957) pro posed anal y sis based on pa ram e ters Q and S. The space (S, Q) is di - vided into two re gions (me an der ing rivers, braided rivers) sep a - rated by lines spec i fied by sim ple equa tions. An other ap proach is the Rosgen clas si fi ca tion (Rosgen and Silvey, 1998; Fig. 1).

This clas si fi ca tion con cerns two ba sic types: sin gle-chan nel and mul ti ple-chan nel rivers. The Rosgen clas si fi ca tion is based on the slope and size of the dom i nant grains of the chan nel ma - te rial. Note, how ever, that for a given slope and grain size there are sev eral pos si ble types of the river ac cord ing to the Rosgen clas si fi ca tion, e.g. A4, F4b, B4, E4b, C4b, and D4b types for the same range of slope (0.02–0.039) and the same grain size

(gravel). In fact, this clas si fi ca tion does not spec ify the con di - tions for form ing any type of river.

How ever, ac cord ing to Leeder (2011: p. 250): “No rigid clas - si fi ca tion of any sin gle-chan nel on any thing lon ger than a reach level (a dozen or so chan nel widths) is uni ver sally prac ti ca ble since many rivers show down stream com bi na tions of sin u os ity and braid ing. How ever, many do not and some ra tio nal dis crim i - na tory clas si fi ca tion may be worth while”. It means that there ex - ists a large group of tran si tional rivers that are not in cluded in the com mon clas si fi ca tion. More over, Leeder (2011: p. 250) stated also: “On the face of it, the mag ni tude of the en ergy avail - able to a stream chan nel and the grain size of the sed i ment sup - plied to it might be con sid ered to be pri mary dy namic vari ables”.

Note, the stream en ergy is given by the for mula:

W = rgQS

where: r – den sity of liq uid, g – grav ity, Q – dis charge, and S – slope.

In the pres ent pa per we used a nu mer i cal model of the river and tried to de ter mine the val ues of the pa ram e ters – dis - charge Q (m3s–1), slope S (rad) and grain size d (mm) – nec es - sary for the for ma tion of a given type of river. There are many pa pers (e.g., Magnuszewski and Gutry-Korycka, 2009a, b) that con cen trate on the prac ti cal as pects e.g., hydrotechnical, flood con trol, and pro tec tion. How ever, ac cord ing to our best knowl edge our pa per is one of the first pre sent ing clas si fi ca - tion for this range of pa ram e ters (rel a tively high slope S and small or me dium dis charge Q) based on nu mer i cal model sim - u la tions.

* Corresponding author, e-mail: Katarzyna.Misiura@fuw.edu.pl Received: May 10, 2016; accepted: October 18, 2016; first published online: December 12, 1016

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PARAMETERS OF CONSIDERED MODELS

The no ta tion of some pa ram e ters of the model used in the pa per is shown in Ta ble 1 and Ap pen dix 1*. We used a rather high slope of the chan nel S – in the range of 0.01–0.04. We in - ves ti gated the be hav iour of a river with such a range for a few rea sons:

– this range is less ex plored;

– we plan in fu ture to use our cal cu la tions for com par i son with rivers on Ti tan, where the typ i cal slope is prob a bly higher (e.g., the range 0.03–0.07 was found by Perron et al., 2006) than for many ter res trial rivers.

The cho sen dis charge Q (30–200 m3 s–1) cor re sponds with small and me dium ter res trial rivers and me dium Titanian rivers (e.g., Jaumann et al., 2008; Burr, 2010). The larg est Titanian dis charge is prob a bly ~1600 m3 s–1. This value is ob tained for an ep i sodic river un der the as sump tion that the flow has short re cur rence in ter vals.

The sizes of grains d of the river bed, bedload, and sus - pended load are the same for a given model and for each model we used grains of one size only. The cal cu la tions are per formed

for the fol low ing grain sizes: 0.1, 1, 10, 100, 200 mm. These sizes are com mon in ter res trial as well as in Titanian rivers (e.g., Leopold and Emmett, 1976; Tomasko et al., 2005). Tak ing grains of one size only is a typ i cal phys i cal ap proach to sim plify the sit u a tion. In nat u ral rivers it cor re sponds to cases where one grain-size dom i nates. The sin gle value (D50) is of ten used as a pa ram e ter char ac ter iz ing the grain size dis tri bu tion (e.g., Yalin, 1992: sec tion 1.5).

The size and ini tial to pog ra phy of the do main of cal cu la tions are given in Fig ure 2. The length of the do main is 1 km and its

Fig. 1. Clas si fi ca tion of rivers ac cord ing to Rosgen and Silvey (1998)

T a b l e 1 Com mon pa ram e ters used for all mod els con sid ered

Pa ram e ter Value

Grav i ta tional ac cel er a tion [m s–2] 9.817

Spe cific grav ity 2.65

Ki ne matic vis cos ity [m2 s–1] 1.52 × 10–6 Sus pended load con cen tra tion [kg m–3] 0.1 Bedload trans port rate [kg m–1s–1] 0.1

* Supplementary data associated with this article can be found, in the online version, at doi: 10.7306/gq.1330

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width is 0.2 km. This width al lows for the de vel op ment of a few par al lel chan nels. Note that the do main is con stant, so lat eral bound aries rep re sent "rigid banks" that can not be eroded. For - tu nately, the mod elled river is rather nar row and the stream (or streams) is usu ally sep a rated from the bound ary of the do main by lat eral bars. These bars could be sub ject to ero sion, so some lat eral mo tion of the river chan nel could be also sim u lated (in the range of the do main).

The ini tial and bound ary con di tions for all mod els are given in Fig ure 2, Ta ble 1 and Ap pen dix 1. The dis charge Q, bedload, and sus pended load are given at the in flow. For the val ues of Q used in our sim u la tion, the flow is ap prox i mately steady and the bed change is rel a tively slow. Us ing the same grain size for the bedload and the sus pended load at the in flow is not a cru cial as - sump tion be cause “nat u ral” dis tri bu tions (i.e. dis tri bu tions cor - re spond ing to cur rent ve loc ity fields and the in ten sity of tur bu - lence) will be es tab lished in a few tens of metres from the be gin -

ning of the sec tion of the river con sid ered. Then the fine frac tion will be come the sus pended load, while the coarse frac tion will be come the bedload.

We mod elled the evo lu tion of the river for at least 100–200 days. This time is long enough to achieve sta bi li za tion (i.e. the ini tial ar ti fi cial shape of the wa ter sur face changes into a nat u ral one af ter a few days). Note also that the evo lu tion for the first 11 days could in clude some tran sient ef fects re sult ing from the ar ti - fi cial ini tial shape of the chan nel. Later, tran sient ef fects are small and the river could be treated as a natural one.

The Man ning co ef fi cient and po ros ity are given in Fig ures 3 and 4. The val ues of the Man ning co ef fi cient for grains up to 1 mm and for some cho sen greater val ues are taken from Melosh (2011, af ter Arcement and Schnei der, 1989) and Arcement and Schnei der (2013). For other larger grains an in - ter po la tion is used. A sim i lar pro ce dure is used for the po ros ity (Fig. 4 based on ta ble 2_1 from McWhorter and Sunada, 1977).

Fig. 2. The ini tial bed to pog ra phy and other in for ma tion about the do main used for our sim u la tions A – to pog ra phy, B – cross sec tion along the river, C – cross-sec tion across the river;

ar rows in di cate places where ini tial or bound ary con di tions are spec i fied

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NUMERICAL MODEL

We chose nu mer i cal mod el ling as our method of in ves ti ga - tion. Of course, even the best nu mer i cal pro gram uses some sim pli fi ca tions. How ever, this method has some ad van tages com pared to field re search and lab o ra tory mod els. Con trary to field re search, we have good con trol of the pa ram e ters (e.g., dis charge). On the other hand, the lab o ra tory mod els are much smaller than real rivers. We used the same nu mer i cal pack age as in our pre vi ous work (Misiura and Czechowski, 2015; Witek and Czechowski, 2015), i.e. CCHE2D pack age. Ba sic in for ma -

tion about this pro gram is given be low. Full in for ma tion is in Wu (2001), Jia and Wang (2001) and Zhang (2006).

In the pack age CCHE2D, flow dy nam ics is de ter mined by so lu tion of the Navier-Stokes equa tions for depth-in te grated, two-di men sional tur bu lent flow and a depth-in te grated con ti nu - ity equa tion (Jia and Wang, 2001). The trans port of sed i ments is mod elled us ing a three-di men sional con vec tion-dif fu sion equa tion. The k–e model of tur bu lence is used (Zhang, 2006).

The pack age uses the fi nite el e ment method. The nu mer i cal mesh for our model con sists of quad ri lat eral fi nite el e ments with 9600 mesh nodes (for de tails see Zhang, 2006). In these mesh nodes the solver gives: ve loc ity vec tor, ve loc ity mag ni tude (i.e.

Fig. 3. Man ning co ef fi cient ver sus size of grains (in ter po la tion is made by a log a rithm func tion rep re sented by black, straight line) Data are ob tained by in ter po la tion of the data from Melosh (2011, af ter Arcement and Schnei der, 1989) and Arcement and Schnei der (2013)

Fig. 4. Po ros ity ver sus size of grains

Data are ob tained by in ter po la tion of the data from McWhorter and Sunada (1977)

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the ab so lute value of the ve loc ity), river depth, bed change (i.e.

the dif fer ence of the cur rent bed to pog ra phy from the ini tial bed to pog ra phy), bedload, and sus pended load. The for mu las for bedload and sus pended load are given in Wu (2001). The von Karman con stant is as sumed to be 0.41 (see also dis cus sion in Frenzen and Vogel, 1994). The ini tial time step is 1 s, how ever, if nec es sary it is re duced for sta bil ity. The pack age is widely used for sci en tific and en gi neer ing ap pli ca tions (e.g., for Vistula River; Wu, 2001; Jia and Wang, 2001; Zhang, 2006; Magnu - sze wski and Gutry-Korycka, 2009a, b).

RESULTS

The ini tial shape of the river val ley (the com pu ta tional do - main) is very sim pli fied (Fig. 2). The ini tial morphologic type of river is de ter mined to some de gree by this ar ti fi cial shape, and there fore can not be used for clas si fi ca tion. The chan nel grad u - ally evolves dur ing the sim u la tion into a “nat u ral” shape. There - fore we used the fol low ing pro ce dure for clas si fi ca tion of our mod elled rivers:

1 – Start ing and con tin u ing the sim u la tion for 11 days. The be hav iour of the river dur ing this pe riod is not used for the clas si fi ca tion of the re sults.

2 – Con tinue sim u la tion for the next 100–240 days. Dur ing this time some sta bi li za tion is achieved in di cat ing that the river could be treated as a nat u ral river.

3 – We used two ap proaches for river clas si fi ca tion:

a. based on the spa tial and tem po rary changes;

b. based on braid-chan nel ra tio B (see def i ni tion of B in Friend and Sinha, 1993).

4 – For the first ap proach, the be hav iour from the 11th day to the end of sim u la tion is used for clas si fi ca tion in the fol - low ing way:

– if the river flows in a sin gle-chan nel for the whole du ra - tion it is clas si fied as a sin gle-chan nel river;

– if the river flows in a sin gle-chan nel in some part of the do main but formed a mul ti ple-chan nel river in an other part of do main or dur ing sim u la tion it changes its type (e.g., ini tially mul ti ple-chan nel river evolves in a sin - gle-chan nel river) it is clas si fied as tran si tional type;

– if the river flows in a few chan nels in the do main it is clas - si fied as a mul ti ple-chan nel river.

5 – For the sec ond ap proach we used the fol low ing pro ce - dure:

a. we cal cu late the co ef fi cient B for the fi nal form of the river (usu ally af ter 240 days of sim u la tion);

b. we de fine three ranges of B:

– B <1.2 – sin gle-chan nel river;

– 1.2£ B <1.7 – tran si tional river;

– B £1.7 – mul ti ple-chan nel river.

These ranges give good char ac ter iza tion of a river.

The re sults of the sim u la tions are pre sented in Fig ures 5–8.

Fig ure 5 gives some ex am ples of the types of river in our re - sults. Note that most of the ex am ples are “tran si tional” rivers rather than pure river types.

DISCUSSION

We found that many of our sim u lated rivers be long to the tran si tional type. In fact this in di cates that our cal cu la tions are re al is tic be cause most of the real rivers are also of tran si tional type (see quo ta tion from Leeder, 2011 in In tro duc tion).

The num ber of tran si tional types de pends on the cho sen num ber of pure types. As sum ing 4 pure types (straight, me an - der ing, braided, anastomosing) then e.g., a slightly sin u ous river could be treated as a tran si tional one be tween straight and me an der ing rivers de pend ing on the pa ram e ter of sin u os ity K.

In our re search we used two pure types: sin gle-chan nel rivers and mul ti ple-chan nel rivers. The rivers are clas si fied us ing pa - ram e ter B (braid-chan nel ra tio) de fined in Friend and Sinha (1993). We use cri te rion 1.2< B <1.7 for tran si tional rivers.

Let us dis cuss Fig ure 6. It gives the po si tions of rivers in space (Q, d) for all con sid ered slopes S (i.e. in fact in 3D space).

We used a re gres sion to de ter mine a power func tion that ap - prox i mates data for tran si tional rivers. For all fig ures we use the same col our code: blue for sin gle-chan nel rivers, red for tran si - tional rivers, and green for mul ti chan nel rivers. The size of the cir cle cor re sponds to the value of S (see leg end in Fig. 6). We found that the trend line for tran si tional rivers is an in creas ing func tion of Q. This is in agree ment with the the ory (e.g., Leeder, 2011) that gives the for mula: S = a db Q–c where a, b, and c are pos i tive num bers. It could be also ex plained in the fol low ing way: the type of river is partly de ter mined by the trans port of sed i ments, so larger d re quires higher ve loc ity and con se - quently larger Q. Fig ure 6 in di cates also that sin gle-chan nel rivers are formed for low Q for all ranges of d and for large Q if d is also large.

The next fig ure (Fig. 7) pres ents our re sults but in 2D space (d/Q, S). As be fore, for tran si tional rivers we cal cu lated the trend line rep re sented by a power func tion. It is also an in creas - ing func tion. This fig ure shows better dif fer ent prop er ties than Fig ure 6. In par tic u lar note that for in creas ing val ues of S the num ber of mul ti chan nel rivers is de creas ing (6 for S = 0.01 and 2 only for S = 0.04).

A com par i son of our re sults with the re sults of Lane (1957) and Leopold and Wolman (1957) is given in Fig ure 8. Note that our re sults for S are sig nif i cantly higher than for S in those pa - pers. In fact we do not know of any other sys tem atic re search of rivers evo lu tion for our range of S. Both our trend line (red line in Fig. 8) and trend lines from Lane (1957) and Leopold and Wolman (1957) are in creas ing power func tions. Note, how ever, the sig nif i cantly lower ab so lute value of the ex po nent in our func tion (our 0.026 com par ing to 0.23 and 0.44).

The cri te rion used for the choice of tran si tional rivers in Fig - ure 8 (with out B) is dif fer ent than the cri te rion used in Fig ures 6 and 7 – see de scrip tion in Re sults. Gen er ally the re sults of both cri te ria are rather sim i lar with some ex cep tion, de scribed above in Fig ure 5. Our lines of trend in Fig ures 6 and 7 are con sis tent with for mula S = a db Q–c af ter Leeder (2011).

CONCLUSIONS

Us ing nu mer i cal sim u la tions we in ves ti gate river evo lu tion for large S, i.e. larger than rivers in ves ti gated by most of other sci en tists (e.g., Lane, 1957).

Sim u la tions for small grains (i.e. d = 0.1 mm) of ten meet nu - mer i cal prob lems (e.g., too large area of dry re gions).

We ob tained three main types of rivers (sin gle-chan nel, mul ti ple-chan nels, and tran si tional). There is also a nice ex am - ple of a braided river (No. 31, S = 0.02).

We used two meth ods for clas si fi ca tion of the rivers (with B cal cu lated for the fi nal form of the river and method with out B but con sid er ing also changes in time). Both meth ods are equiv - a lent for most cases.

We found that the trend line for tran si tional rivers is an in - creas ing func tion of Q in space (Q, d). This is in agree ment with

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S = 0.02, No. 31, B = 2.31 – multiple channel river

S = 0.03, No. 25, B = 1.87 – multiple channel river

S = 0.04, No. 21, B = 1.2 – a transitional type. It is a single channel river in a part of the domain but forms multiple channels in another part of domain

S = 0.04, No. 15, B = 1.77– a transitional type. It is finally single channel river but evolves from a multiple channel river

Bed change [m]

Bed change [m]

Bed change [m]

Bed change [m]

Bed change [m]

S = 0.02, No. 15, B = 1.03 – a single channel river

Fig. 5. Some ex am ples of the types of river in our re sults

Dif fer ent col our scales are used for dif fer ent pan els; note that nr 15 for S = 0.04 is clas si fied as tran si tional river type ac cord ing to cri te rion a), but also as a mul ti ple-chan nel river ac cord ing to cri te rion b)

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Fig. 6. Po si tions of ob tained rivers in space (Q, d) for all con sid ered slopes S and trend line for tran si tional rivers

Only cir cles cor re spond ing to tran si tional rivers (i.e. red) are used for a re gres sion to de ter mine power func tion; note that at any given point (Q, d) there could be a few re sults cor re spond ing to dif fer ent val ues of S; the power func tion ob tained in creases with Q ac cord ing to pre dic tion of the ory (e.g., Leeder, 2011);

we di vide all rivers based on pa ram e ter B (dif fer ent colours) and on slope (dif fer ent sizes of cir cles) – see leg end next to the graph

Fig. 7. Po si tions of ob tained rivers in space (d/Q, S) for all con sid ered slopes S and trend line cal cu lated for tran si tional rivers

Only cir cles cor re spond ing to tran si tional rivers (i.e. red) are used for a re gres sion to de ter mine power func tion; note that the func tion ob tained in creases with Q ac cord ing to pre dic tion of the ory

(e.g., Leeder, 2011); we di vide all rivers based on pa ram e ter B

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for mula: S = a db Q–c. Fig ure 6 in di cates also com mon sin - gle-chan nel rivers for small Q or for large Q if d is also high.

For large val ues of S the num ber of mul ti chan nel rivers de - creases (Fig. 7).

The ex po nent in power func tion for the trend line for large val ues of S is sig nif i cantly lower than for low S (Fig. 8).

Ac knowl edge ments. Pro grams de vel oped by the Na tional Cen ter for Com pu ta tional Hydroscience and En gi neer ing at The Uni ver sity of Mis sis sippi were used in this re search. We are grate ful to Prof. A. Wysocka from Fac ulty of Ge ol ogy at Uni ver - sity of War saw. The au thors would like to thank the anon y mous re view ers for their re marks and valu able com ments.

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