\begin{thebibliography}{84}
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\bibitem[{\textit{Ashida and Michiue}(1972)}]{Ashida72}
Ashida, K., and M.~Michiue (1972), Study on hydraulic resistance and bed-load
  transport rate in alluvial streams, \textit{J. Soc. Civ. Eng. Trans.},
  \textit{206}, 59--69.

\bibitem[{\textit{Ashida et~al.}(1990)\textit{Ashida, Egashira, Liu, and
  Umemoto}}]{Ashida90}
Ashida, K., S.~Egashira, B.~Liu, and M.~Umemoto (1990), Sorting and bed
  topography in meander channels, \textit{Annuals Disaster Prevention Institute
  Kyoto University}, \textit{33 B-2}, 261--279, (in Japanese).

\bibitem[{\textit{Banks et~al.}(1992)\textit{Banks, Brooks, Cairns, Davis, and
  Stacey}}]{Banks92}
Banks, J., J.~Brooks, G.~Cairns, G.~Davis, and P.~Stacey (1992), On devaney's
  definition of chaos, \textit{The American Mathematical Monthly},
  \textit{99}(4), 332--334, \doi{10.2307/2324899}.

\bibitem[{\textit{{\van{Bendegom}{Van}{van}}~Bendegom}(1947)}]{vanBendegom47}
{\van{Bendegom}{Van}{van}}~Bendegom, L. (1947), Eenige beschouwingen over
  riviermorphofogie en rivierverbetering, \textit{De Ingenieur},
  \textit{59}(4), 1--11, (in Dutch).

\bibitem[{\textit{Blom}(2008)}]{Blom08}
Blom, A. (2008), Different approaches to handling vertical and streamwise
  sorting in modeling river morphodynamics, \textit{Water Resour. Res.},
  \textit{44}(3), {W03,415}, \doi{10.1029/2006WR005474}.

\bibitem[{\textit{Blom et~al.}(2006)\textit{Blom, Parker, Ribberink, and
  de~Vriend}}]{Blom06_2}
Blom, A., G.~Parker, J.~S. Ribberink, and H.~J. de~Vriend (2006), Vertical
  sorting and the morphodynamics of bed-form-dominated rivers: An equilibrium
  sorting model, \textit{J. Geophys. Res., Earth Surface}, \textit{111},
  F01,006, \doi{10.1029/2004JF000175}.

\bibitem[{\textit{Blom et~al.}(2008)\textit{Blom, Ribberink, and
  Parker}}]{Blom08_2}
Blom, A., J.~S. Ribberink, and G.~Parker (2008), Vertical sorting and the
  morphodynamics of bed form-dominated rivers: A sorting evolution model,
  \textit{J. Geophys. Res., Earth Surface}, \textit{113}, F01,019,
  \doi{10.1029/2006JF000618}.

\bibitem[{\textit{Blom et~al.}(2016)\textit{Blom, Viparelli, and
  Chavarr\'ias}}]{Blom16}
Blom, A., E.~Viparelli, and V.~Chavarr\'ias (2016), The graded alluvial river:
  Profile concavity and downstream fining, \textit{Geophys. Res. Lett.},
  \textit{43}, 1--9, \doi{10.1002/2016GL068898}.

\bibitem[{\textit{Blom et~al.}(2017)\textit{Blom, Chavarr\'ias, Ferguson, and
  Viparelli}}]{Blom17}
Blom, A., V.~Chavarr\'ias, R.~I. Ferguson, and E.~Viparelli (2017), Advance,
  retreat, and halt of abrupt gravel-sand transitions in alluvial rivers,
  \textit{Geophys. Res. Lett.}, \doi{10.1002/2017GL074231}.

\bibitem[{\textit{Chavarr\'ias and Ottevanger}(2016)}]{Chavarrias16}
Chavarr\'ias, V., and W.~Ottevanger (2016), Mathematical analysis of the
  well-posedness of the {H}irano active layer concept in 2{D} models,
  \textit{Tech. Rep. 1230044}, Deltares.

\bibitem[{\textit{Chavarr\'ias et~al.}(2018)\textit{Chavarr\'ias, Stecca, and
  Blom}}]{Chavarrias18}
Chavarr\'ias, V., G.~Stecca, and A.~Blom (2018), Ill-posedness in modelling
  mixed-sediment river morphodynamics, \textit{Adv. Water Resour.}, (in press).

\bibitem[{\textit{Colombini et~al.}(1987)\textit{Colombini, Seminara, and
  Tubino}}]{Colombini87}
Colombini, M., G.~Seminara, and M.~Tubino (1987), Finite-amplitude alternate
  bars, \textit{J. Fluid Mech.}, \textit{181}, 213--232,
  \doi{10.1017/S0022112087002064}.

\bibitem[{\textit{Courant et~al.}(1928)\textit{Courant, Friedrichs, and
  Lewy}}]{Courant28}
Courant, R., K.~Friedrichs, and H.~Lewy (1928), \"uber die partiellen
  differenzengleichungen der mathematischen physik, \textit{Mathematische
  Annalen}, \textit{100}, 32--74.

\bibitem[{\textit{Darmofal and Schmid}(1996)}]{Darmofal96}
Darmofal, D., and P.~Schmid (1996), The importance of eigenvectors for local
  preconditioners of the euler equations, \textit{J. Comput. Phys.},
  \textit{127}(2), 346--362, \doi{10.1006/jcph.1996.0180}.

\bibitem[{\textit{Devaney}(1989)}]{Devaney89}
Devaney, R.~L. (1989), \textit{An Introduction to Chaotic Dynamical Systems},
  360 pp., Westview Press.

\bibitem[{\textit{Duan}(1998)}]{Duan98}
Duan, G.~J. (1998), Simulation of alluvial channel migration processes with a
  two-dimensional numerical model, Ph.D. thesis, Cener for Computational
  Hydroscience and Engineering, The University of Mississippi.

\bibitem[{\textit{Duan}(2004)}]{Duan04}
Duan, J.~G. (2004), Simulation of flow and mass dispersion in meandering
  channels, \textit{J. Hydraul. Eng.}, \textit{130}(10), 964--976,
  \doi{10.1061/(ASCE)0733-9429(2004)130:10(964)}.

\bibitem[{\textit{Duan et~al.}(2001)\textit{Duan, Wang, and Jia}}]{Duan01}
Duan, J.~G., S.~S.~Y. Wang, and Y.~Jia (2001), The applications of the enhanced
  {CCHE2D} model to study the alluvial channel migration processes, \textit{J.
  Hydraul. Res.}, \textit{39}(5), 469--480,
  \doi{10.1080/00221686.2001.9628272}.

\bibitem[{\textit{Dulal et~al.}(2010)\textit{Dulal, Kobayashi, Shimizu, and
  Parker}}]{Dulal10}
Dulal, K.~P., K.~Kobayashi, Y.~Shimizu, and G.~Parker (2010), Numerical
  computation of free meandering channels with the application of slump blocks
  on the outer bends, \textit{J. Hydro-environ. Res.}, \textit{3}(4), 239--246,
  \doi{10.1016/j.jher.2009.10.012}.

\bibitem[{\textit{Egiazaroff}(1965)}]{Egiazaroff65}
Egiazaroff, I.~V. (1965), Calculation of nonuniform sediment concentrations,
  \textit{J. Hydraulics Div.}, \textit{91}(4), 225--247.

\bibitem[{\textit{Elder}(1959)}]{Elder59}
Elder, J.~W. (1959), The dispersion of marked fluid in turbulent shear flow,
  \textit{J. Fluid Mech.}, \textit{5}(4), 544--560,
  \doi{10.1017/S0022112059000374}.

\bibitem[{\textit{Engelund}(1974)}]{Engelund74}
Engelund, F. (1974), Flow and bed topography in channel bends, \textit{J.
  Hydraulics Div.}, \textit{100}(11), 1631--1648.

\bibitem[{\textit{Engelund and Hansen}(1967)}]{Engelund67}
Engelund, F., and E.~Hansen (1967), Monograph on sediment transport in alluvial
  streams, \textit{{T}ech. {R}ep.}, Hydraul. Lab., Tech. Univ. of Denmark,
  Copenhagen, Denmark., 63 pp.

\bibitem[{\textit{Engelund and Skovgaard}(1973)}]{Engelund73}
Engelund, F., and O.~Skovgaard (1973), On the origin of meandering and braiding
  in alluvial streams, \textit{J. Fluid Mech.}, \textit{57}(2), 289--302,
  \doi{10.1017/S0022112073001163}.

\bibitem[{\textit{Erdogan and Chatwin}(1967)}]{Erdogan67}
Erdogan, M.~E., and P.~C. Chatwin (1967), The effects of curvature and buoyancy
  on the laminar dispersion of solute in a horizontal tube, \textit{J. Fluid
  Mech.}, \textit{29}(3), 465--484, \doi{10.1017/S0022112067000977}.

\bibitem[{\textit{Exner}(1920)}]{Exner20}
Exner, F.~M. (1920), Zur {P}hysik der {D}{\"u}nen, \textit{Akad. Wiss. Wien
  Math. Naturwiss}, \textit{129(2a)}, 929--952, (in German).

\bibitem[{\textit{Fischer}(1967)}]{Fischer67}
Fischer, H.~B. (1967), The mechanics of dispersion in natural streams,
  \textit{J. Hydraulics Div.}, \textit{96}(6), 187--216.

\bibitem[{\textit{Fischer}(1969)}]{Fischer69}
Fischer, H.~B. (1969), The effect of bends on dispersion in streams,
  \textit{Water Resour. Res.}, \textit{5}(2), 496--506,
  \doi{10.1029/WR005i002p00496}.

\bibitem[{\textit{Fischer}(1973)}]{Fischer73}
Fischer, H.~B. (1973), Longitudinal dispersion and turbulent mixing in
  open-channel flow, \textit{Annu. Rev. Fluid Mech.}, \textit{5}(1), 59--78,
  \doi{10.1146/annurev.fl.05.010173.000423}.

\bibitem[{\textit{Flokstra}(1977)}]{Flokstra77}
Flokstra, C. (1977), The closure problem for depth-averaged 2-d flow,
  \textit{Proc. 18th Congress of IAHR, Baden-Baden, Germany, 15-19 August}.

\bibitem[{\textit{Freds\oe}(1978)}]{Fredsoe78}
Freds\oe, J. (1978), Meandering and braiding of rivers, \textit{J. Fluid
  Mech.}, \textit{84}(4), 609--624, \doi{10.1017/S0022112078000373}.

\bibitem[{\textit{Hadamard}(1923)}]{Hadamard22}
Hadamard, J.~S. (1923), \textit{Lectures on {C}auchy's problem in linear
  partial differential equations}, 316 pp., {Y}ale {U}niversity Press, New
  Haven.

\bibitem[{\textit{Hasegawa}(1983)}]{Hasegawa83}
Hasegawa, K. (1983), Hydraulic study on the plane and riverbed shape and flow
  of alluvial meander, Ph.D. thesis, Hokkaido University, (in Japanese).

\bibitem[{\textit{Hirano}(1971)}]{Hirano71}
Hirano, M. (1971), River bed degradation with armoring, \textit{Trans. Jpn.
  Soc. Civ. Eng}, \textit{195}, 55--65, \doi{10.2208/jscej1969.1971.195\_55}.

\bibitem[{\textit{Hoey and Ferguson}(1994)}]{Hoey94}
Hoey, T.~B., and R.~Ferguson (1994), Numerical simulation of downstream fining
  by selective transport in gravel bed rivers: Model development and
  illustration, \textit{Water Resour. Res.}, \textit{30}(7), 2251--2260,
  \doi{10.1029/94WR00556}.

\bibitem[{\textit{Iwagaki}(1956)}]{Iwagaki56}
Iwagaki, Y. (1956), Hydrodynamical study on critical tractive force,
  \textit{Trans. of JSCE}, \textit{41}.

\bibitem[{\textit{Iwasaki et~al.}(2017)\textit{Iwasaki, Nelson, Shimizu, and
  Parker}}]{Iwasaki17}
Iwasaki, T., J.~Nelson, Y.~Shimizu, and G.~Parker (2017), Numerical simulation
  of large-scale bed load particle tracer advection-dispersion in rivers with
  free bars, \textit{J. Geophys. Res., Earth Surface}, \textit{122}(4),
  847--874, \doi{10.1002/2016JF003951}, 2016JF003951.

\bibitem[{\textit{Jagers}(2003)}]{Jagers03_PhD}
Jagers, B. (2003), Modelling planform changes of braided rivers, Ph.D. thesis,
  Universiteit Twente.

\bibitem[{\textit{Jia and Wang}(1999)}]{Jia99}
Jia, Y., and S.~S.~Y. Wang (1999), Numerical model for channel flow and
  morphological change studies, \textit{J. Hydraul. Eng.}, \textit{125}(9),
  924--933, \doi{10.1061/(ASCE)0733-9429(1999)125:9(924)}.

\bibitem[{\textit{Joseph and Saut}(1990)}]{Joseph90}
Joseph, D., and J.~Saut (1990), Short-wave instabilities and ill-posed
  initial-value problems, \textit{Theor. Comput. Fluid Mech.}, \textit{1}(4),
  191--227, \doi{10.1007/BF00418002}.

\bibitem[{\textit{Kabanikhin}(2008)}]{Kabanikhin08}
Kabanikhin, S.~I. (2008), Definitions and examples of inverse and ill-posed
  problems, \textit{J. Inv. Ill-Posed Problems}, \textit{16}, 317--357,
  \doi{10.1515/JIIP.2008.019}.

\bibitem[{\textit{Kalkwijk and Booij}(1986)}]{Kalkwijk86}
Kalkwijk, J. P.~T., and R.~Booij (1986), Adaptation of secondary flow in
  nearly-horizontal flow, \textit{J. Hydraul. Res.}, \textit{24}(1), 19--37,
  \doi{10.1080/00221688609499330}.

\bibitem[{\textit{Kalkwijk and Vriend}(1980)}]{Kalkwijk80}
Kalkwijk, J. P.~T., and H.~J.~D. Vriend (1980), Computation of the flow in
  shallow river bends, \textit{J. Hydraul. Res.}, \textit{18}(4), 327--342,
  \doi{10.1080/00221688009499539}.

\bibitem[{\textit{Kassem and Chaudhry}(2002)}]{Kassem02}
Kassem, A.~A., and M.~H. Chaudhry (2002), Numerical modeling of bed evolution
  in channel bends, \textit{J. Hydraul. Eng.}, \textit{128}(5), 507--514,
  \doi{10.1061/(ASCE)0733-9429(2002)128:5(507)}.

\bibitem[{\textit{Kelvin}(1871)}]{Kelvin71}
Kelvin, L. S. W. T. F.~. (1871), Hydrokinetic solutions and observations,
  \textit{Philos. Mag.}, \textit{42}(281), 362--377,
  \doi{10.1080/14786447108640585}.

\bibitem[{\textit{Koch and Flokstra}(1980)}]{Koch80}
Koch, F.~G., and C.~Flokstra (1980), Bed level computations for curved alluvial
  channels, \textit{Tech. Rep. 240}, Delft Hydraulics Laboratory, The
  Netherlands.

\bibitem[{\textit{Kuroki and Kishi}(1984)}]{Kuroki84}
Kuroki, M., and T.~Kishi (1984), Regime criteria on bars and braids in alluvial
  straight channels, \textit{Proceedings of the Japanese Society of Civil
  Engineers}, \textit{342}, 87--96, (in Japanese).

\bibitem[{\textit{Lanzoni and Tubino}(1999)}]{Lanzoni99}
Lanzoni, S., and M.~Tubino (1999), Grain sorting and bar instability,
  \textit{J. Fluid Mech.}, \textit{393}, 149--174,
  \doi{10.1017/S0022112099005583}.

\bibitem[{\textit{Lanzoni et~al.}(2006)\textit{Lanzoni, Siviglia, Frascati, and
  Seminara}}]{Lanzoni06}
Lanzoni, S., A.~Siviglia, A.~Frascati, and G.~Seminara (2006), Long waves in
  erodible channels and morphodynamic influence, \textit{Water Resour. Res.},
  \textit{42}(6), \doi{10.1029/2006WR004916}.

\bibitem[{\textit{Latteux}(1995)}]{Latteux95}
Latteux, B. (1995), Techniques for long-term morphological simulation under
  tidal action, \textit{Mar. Geol.}, \textit{126}(126), 129--141,
  \doi{10.1016/0025-3227(95)00069-B}.

\bibitem[{\textit{Legleiter and Kyriakidis}(2006)}]{Legleiter06}
Legleiter, C.~J., and P.~C. Kyriakidis (2006), Forward and inverse
  transformations between cartesian and channel-fitted coordinate systems for
  meandering rivers, \textit{Math. Geol.}, \textit{38}(8), 927--958,
  \doi{10.1007/s11004-006-9056-6}.

\bibitem[{\textit{Madsen et~al.}(1988)\textit{Madsen, Rugbjerg, and
  Warren}}]{Madsen88}
Madsen, P., M.~Rugbjerg, and I.~Warren (1988), Subgrid modelling in depth
  integrated flows, \textit{21st International Conference on Coastal
  Engineering}.

\bibitem[{\textit{Meyer-Peter and M{\"u}ller}(1948)}]{MeyerPeter48}
Meyer-Peter, E., and R.~M{\"u}ller (1948), Formulas for bed-load transport, in
  \textit{Proc. 2nd Meeting Int. Assoc. Hydraul. Struct. Res.}, pp. 39--64,
  Stockholm.

\bibitem[{\textit{Olesen}(1982)}]{Olesen82}
Olesen, K.~W. (1982), Influence of secondary flow on meandering rivers,
  \textit{Tech. Rep. 1-82}, Laboratory of Fluid Mechanics, Delft University of
  Technology.

\bibitem[{\textit{Olesen}(1983)}]{Olesen83}
Olesen, K.~W. (1983), Alternate bars in and meandering of alluvial rivers,
  \textit{Tech. Rep. 83-1}, WL Delft Hydraulics.

\bibitem[{\textit{Ottevanger et~al.}(2015)\textit{Ottevanger, Giri, and
  Sloff}}]{Ottevanger15}
Ottevanger, W., S.~Giri, and K.~Sloff (2015), Sustainable fairway rhinedelta
  ii, \textit{Tech. rep.}, Deltares.

\bibitem[{\textit{Parker}(1976)}]{Parker76}
Parker, G. (1976), On the cause and characteristic scales of meandering and
  braiding in rivers, \textit{J. Fluid Mech.}, \textit{76}(3), 457--480,
  \doi{10.1017/S0022112076000748}.

\bibitem[{\textit{Parker}(1991)}]{Parker91_1}
Parker, G. (1991), Selective sorting and abrasion of river gravel. {I}: Theory,
  \textit{J. Hydraul. Eng.}, \textit{117}(2), 131--147,
  \doi{10.1061/(ASCE)0733-9429(1991)117:2(131)}.

\bibitem[{\textit{Parker and Klingeman}(1982)}]{Parker82}
Parker, G., and P.~C. Klingeman (1982), On why gravel bed streams are paved,
  \textit{Water Resour. Res.}, \textit{18}(5), 1409--1423,
  \doi{10.1029/WR018i005p01409}.

\bibitem[{\textit{Parker et~al.}(1982)\textit{Parker, Klingeman, and
  McLean}}]{Parker82_3}
Parker, G., P.~C. Klingeman, and D.~G. McLean (1982), Bedload and size
  distribution in paved gravel-bed streams, \textit{J. Hydraulics Div.},
  \textit{108}(4), 544--571.

\bibitem[{\textit{Ranasinghe et~al.}(2011)\textit{Ranasinghe, Swinkels,
  Luijendijk, Roelvink, Bosboom, Stive, and Walstra}}]{Ranasinghe11}
Ranasinghe, R., C.~Swinkels, A.~Luijendijk, D.~Roelvink, J.~Bosboom, M.~Stive,
  and D.~Walstra (2011), Morphodynamic upscaling with the {MORFAC} approach:
  Dependencies and sensitivities, \textit{Coastal Eng.}, \textit{58}(8),
  806--811, \doi{10.1016/j.coastaleng.2011.03.010}.

\bibitem[{\textit{Ribberink}(1987)}]{Ribberink87_PhD}
Ribberink, J.~S. (1987), Mathematical modelling of one-dimensional
  morphological changes in rivers with non-uniform sediment, Ph.D. thesis,
  Delft University of Technology, The Netherlands.

\bibitem[{\textit{Rodi}(1982)}]{Rodi82}
Rodi, W. (1982), \textit{Turbulence model and their application in hydraulics},
  IAHR Monographs, 124 pp., {CRC} press.

\bibitem[{\textit{Roelvink}(2006)}]{Roelvink06}
Roelvink, J. (2006), Coastal morphodynamic evolution techniques,
  \textit{Coastal Eng.}, \textit{53}, 277--287,
  \doi{10.1016/j.coastaleng.2005.10.015}.

\bibitem[{\textit{Rozovskii}(1957)}]{Rozovskii57}
Rozovskii, I.~L. (1957), \textit{Flow of water in bends in open channels}, 233
  pp., Academy of Sciences of the Ukrainian SSR, (in English translated by Y.
  Prushansky in 1961).

\bibitem[{\textit{Saint-Venant}(1871)}]{SaintVenant71}
Saint-Venant, A. J. C.~B. (1871), Th\'{e}orie du mouvement non permanent des
  eaux, avec application aux crues des rivi\`{e}res et \`{a} l'introduction des
  mar\'{e}es dans leur lit, \textit{Comptes Rendus des s\'{e}ances de
  l'Acad\'{e}mie des Sciences}, \textit{73}, 237--240, (in French).

\bibitem[{\textit{Schielen et~al.}(1993)\textit{Schielen, Doelman, and
  de~Swart}}]{Schielen93}
Schielen, R., A.~Doelman, and H.~E. de~Swart (1993), On the nonlinear dynamics
  of free bars in straight channels, \textit{J. Fluid Mech.}, \textit{252},
  325--356, \doi{10.1017/S0022112093003787}.

\bibitem[{\textit{Sekine and Parker}(1992)}]{Sekine92_1}
Sekine, M., and G.~Parker (1992), Bed-load transport on transverse slope. i,
  \textit{J. Hydraul. Eng.}, \textit{118}(4), 513--535,
  \doi{10.1061/(ASCE)0733-9429(1992)118:4(513)}.

\bibitem[{\textit{Seminara et~al.}(1996)\textit{Seminara, Colombini, and
  Parker}}]{Seminara96}
Seminara, G., M.~Colombini, and G.~Parker (1996), Nearly pure sorting waves and
  formation of bedload sheets, \textit{J. Fluid Mech.}, \textit{312}, 253--278,
  \doi{10.1017/S0022112096001991}.

\bibitem[{\textit{Sieben}(1994)}]{sieben94}
Sieben, A. (1994), Notes on the mathematical modelling of alluvial mountain
  rivers with graded sediment, \textit{Tech. Rep. 94--3}, Delft University of
  Technology, The Netherlands.

\bibitem[{\textit{Sieben}(1997)}]{Sieben97_PhD}
Sieben, J. (1997), Modelling of hydraulics and morphology in mountain rivers,
  Ph.D. thesis, Delft University of Technology.

\bibitem[{\textit{Siviglia et~al.}(2013)\textit{Siviglia, Stecca, Vanzo,
  Zolezzi, Toro, and Tubino}}]{Siviglia13}
Siviglia, A., G.~Stecca, D.~Vanzo, G.~Zolezzi, E.~F. Toro, and M.~Tubino
  (2013), Numerical modelling of two-dimensional morphodynamics with
  applications to river bars and bifurcations, \textit{Adv. Water Resour.},
  \textit{52}, 243--260, \doi{10.1016/j.advwatres.2012.11.010}.

\bibitem[{\textit{Sloff}(1992)}]{Sloff92}
Sloff, C.~J. (1992), The method of characteristics applied to analyse 2{DH}
  models, \textit{Tech. Rep. 92--4}, Delft University of Technology, The
  Netherlands.

\bibitem[{\textit{Sloff}(1993)}]{Sloff93}
Sloff, K. (1993), Analysis of basic equations for sediment-laden flows,
  \textit{techreport 93-8}, Delft University of Technology, The Netherlands, 54
  pp.

\bibitem[{\textit{Stecca et~al.}(2014)\textit{Stecca, Siviglia, and
  Blom}}]{Stecca14}
Stecca, G., A.~Siviglia, and A.~Blom (2014), Mathematical analysis of the
  {S}aint-{V}enant-{H}irano model for mixed-sediment morphodynamics,
  \textit{Water Resour. Res.}, \textit{50}, 7563--7589,
  \doi{10.1002/2014WR015251}.

\bibitem[{\textit{Sternberg}(1875)}]{Sternberg75}
Sternberg, H. (1875), Untersuchungen {\"u}ber {L}{\"a}ngen- und {Q}uerprofil
  geschiebef{\"u}hrender {F}l{\"u}sse, \textit{{Z}eitschrift f{\"u}r
  {B}auwesen}, \textit{25}, 483--506, (in {G}erman).

\bibitem[{\textit{Struiksma}(1985)}]{Struiksma85_2}
Struiksma, N. (1985), Prediction of 2d bed topography in rivers, \textit{J.
  Hydraul. Eng.}, \textit{111}(8), 1169--1182,
  \doi{10.1061/(ASCE)0733-9429(1985)111:8(1169)}.

\bibitem[{\textit{Talmon et~al.}(1995)\textit{Talmon, Struiksma, and
  Mierlo}}]{Talmon95}
Talmon, A., N.~Struiksma, and M.~V. Mierlo (1995), Laboratory measurements of
  the direction of sediment transport on transverse alluvial-bed slopes,
  \textit{Journal of Hydraulic Research}, \textit{33}(4),
  \doi{10.1080/00221689509498657}.

\bibitem[{\textit{Thomson}(1876)}]{Thomson76}
Thomson, J. (1876), On the origin of windings of rivers in alluvial plains,
  with remarks on the flow of water round bends in pipes, \textit{Proc. Roy.
  Soc. London}, \textit{25}(171-178), 5--8, \doi{10.1098/rspl.1876.0004}.

\bibitem[{\textit{Tubino et~al.}(1999)\textit{Tubino, Repetto, and
  Zolezzi}}]{Tubino99}
Tubino, M., R.~Repetto, and G.~Zolezzi (1999), Free bars in rivers, \textit{J
  Hydraul Res}, \textit{37}(6), 759--775, \doi{10.1080/00221689909498510}.

\bibitem[{\textit{Viparelli et~al.}(2017)\textit{Viparelli, Moreira, and
  Blom}}]{Viparelli17}
Viparelli, E., R.~R.~H. Moreira, and A.~Blom (2017), \textit{Gravel-Bed Rivers:
  Process and Disasters}, chap. 23: Modelling stratigraphy-based {GBR}
  morphodynamics., pp. 609--637, Wiley-Blackwell,
  \doi{10.1002/9781118971437.ch23}.

\bibitem[{\textit{{\van{Vriend}{De}{de}}~Vriend}(1981)}]{deVriend81_PhD}
{\van{Vriend}{De}{de}}~Vriend, H.~J. (1981), Steady flow in shallow channel
  bends, Ph.D. thesis, Delft University of Technology.

\bibitem[{\textit{Yatsu}(1955)}]{Yatsu55}
Yatsu, E. (1955), On the longitudinal profile of the graded river, \textit{EOS,
  Trans. Am. Geophys. Union}, \textit{36(4)}, 655--663.

\bibitem[{\textit{Zimmerman and Kennedy}(1978)}]{Zimmerman78}
Zimmerman, C., and J.~F. Kennedy (1978), Transverse bed slopes in curved
  alluvial streams, \textit{J. Hydraulics Div.}, \textit{104}(1), 33--48.

\end{thebibliography}