\contentsline {figure}{\numberline {1}{\ignorespaces Domain of linear stability of the reference situation (Table \ref {tab:L_ph}) in terms of: (\textbf {a}) wavenumber, and (\textbf {b}) wave length. The continuous line is the separatrix for the case without secondary flow while the dashed line is the separatrix considering secondary flow and a diffusion coefficient $D_H=\SI {5}{m^2/s}$. The circle and square mark the conditions of the numerical simulations (Table \ref {tab:L_num_diff}).}}{18}{figure.1}
\contentsline {figure}{\numberline {2}{\ignorespaces Evolution of the flow depth with time at $y=B/2$ for a location in which initially we find the trough of a bar (i.e., maximum flow depth) for simulation (\textbf {a}) L1, (\textbf {b}) L2, and (\textbf {c}) L3 (Tables \ref {tab:L_num_eq} and \ref {tab:L_num_diff}).}}{19}{figure.2}
\contentsline {figure}{\numberline {3}{\ignorespaces Wave growth domain for the case S1 (Tables \ref {tab:L_ph} and \ref {tab:s_var}) as a function of the wavenumber (\textbf {a}) and wavelength (\textbf {b}). In the green area the growth rate is negative (dampening) while in the red area it is positive (growth). This situation is ill-posed (Section \ref {subsec:def_ill}).}}{21}{figure.3}
\contentsline {figure}{\numberline {4}{\ignorespaces Minimum diffusion coefficient for the case S1 (Tables \ref {tab:L_ph} and \ref {tab:s_var}) to obtain a well-posed model as a function the wavenumber (\textbf {a}) and wavelength (\textbf {b}).}}{22}{figure.4}
\contentsline {figure}{\numberline {5}{\ignorespaces Wave growth domain for the case S3 (Tables \ref {tab:L_ph} and \ref {tab:s_var}) as a function of the wavenumber (\textbf {a}) and wavelength (\textbf {b}). In the green area the growth rate is negative (dampening) while in the red area it is positive (growth). This situation is ill-posed (Section \ref {subsec:def_ill}).}}{23}{figure.5}
\contentsline {figure}{\numberline {6}{\ignorespaces Wave growth domain for the case S4 (Tables \ref {tab:L_ph} and \ref {tab:s_var}) as a function of the wavenumber (\textbf {a}) and wavelength (\textbf {b}). In the green area the growth rate is negative (dampening) while in the red area it is positive (growth). This situation is well-posed (Section \ref {subsec:def_ill}).}}{23}{figure.6}
\contentsline {figure}{\numberline {7}{\ignorespaces Wave growth domain for the case S2 (Tables \ref {tab:L_ph} and \ref {tab:s_var}) as a function of the wavenumber (\textbf {a}) and wavelength (\textbf {b}). In the green area the growth rate is negative (dampening) while in the red area it is positive (growth). This situation is ill-posed (Section \ref {subsec:def_ill}).}}{24}{figure.7}
\contentsline {figure}{\numberline {8}{\ignorespaces Wave growth domain for the case S5 (Tables \ref {tab:L_ph} and \ref {tab:s_var}) as a function of the wavenumber (\textbf {a}) and wavelength (\textbf {b}). In the green area the growth rate is negative (dampening) while in the red area it is positive (growth). This situation is well-posed (Section \ref {subsec:def_ill}).}}{24}{figure.8}
\contentsline {figure}{\numberline {9}{\ignorespaces Flow depth after \SI {15}{s} in the simulation of Case S1.}}{25}{figure.9}
\contentsline {figure}{\numberline {10}{\ignorespaces Flow depth after \SI {60}{s} in the simulation of Case S1.}}{25}{figure.10}
\contentsline {figure}{\numberline {11}{\ignorespaces Flow depth after \SI {60}{s} in the simulation of Case S4.}}{25}{figure.11}
\contentsline {figure}{\numberline {12}{\ignorespaces Wave growth domain for the case B1 (Tables \ref {tab:L_ph} and \ref {tab:B_var}) as a function of the wavenumber (\textbf {a}) and wavelength (\textbf {b}). In the green area the growth rate is negative (dampening) while in the red area it is positive (growth). This situation is well-posed (Section \ref {subsec:def_ill}).}}{26}{figure.12}
\contentsline {figure}{\numberline {13}{\ignorespaces Wave growth domain for the case B2 (Tables \ref {tab:L_ph} and \ref {tab:B_var}) as a function of the wavenumber (\textbf {a}) and wavelength (\textbf {b}). In the green area the growth rate is negative (dampening) while in the red area it is positive (growth). This situation is ill-posed (Section \ref {subsec:def_ill}).}}{27}{figure.13}
\contentsline {figure}{\numberline {14}{\ignorespaces Flow depth at the end of the simulation of Case B2.}}{27}{figure.14}
\contentsline {figure}{\numberline {15}{\ignorespaces Longitudinal profile of the flow depth at the end of the simulation of Case B2.}}{28}{figure.15}
\contentsline {figure}{\numberline {16}{\ignorespaces Flow depth at the end of the simulation of Case B1.}}{28}{figure.16}
\contentsline {figure}{\numberline {17}{\ignorespaces Longitudinal profile of the flow depth at the end of the simulation of Case B1.}}{29}{figure.17}
\contentsline {figure}{\numberline {18}{\ignorespaces Wave growth domain for the case M1 (Tables \ref {tab:ref_mixed} and \ref {tab:mixed_var}) as a function of the wavenumber (\textbf {a}) and wavelength (\textbf {b}). In the green area the growth rate is negative (dampening) while in the red area it is positive (growth). This situation is well-posed (Section \ref {subsec:def_ill}).}}{30}{figure.18}
\contentsline {figure}{\numberline {19}{\ignorespaces Wave growth domain for the case M2 (Tables \ref {tab:ref_mixed} and \ref {tab:mixed_var}) as a function of the wavenumber (\textbf {a}) and wavelength (\textbf {b}). In the green area the growth rate is negative (dampening) while in the red area it is positive (growth). This situation is ill-posed (Section \ref {subsec:def_ill}).}}{31}{figure.19}
\contentsline {figure}{\numberline {20}{\ignorespaces Wave growth domain for the case M3 (Tables \ref {tab:ref_mixed} and \ref {tab:mixed_var}) as a function of the wavenumber (\textbf {a}) and wavelength (\textbf {b}). In the green area the growth rate is negative (dampening) while in the red area it is positive (growth). This situation is ill-posed (Section \ref {subsec:def_ill}).}}{32}{figure.20}
\contentsline {figure}{\numberline {21}{\ignorespaces Longitudinal profile at the center of the domain ($y$=\SI {5}{m}) at the initial state of the simulation of Case I4.}}{37}{figure.21}
\contentsline {figure}{\numberline {22}{\ignorespaces Volume fraction content at the first substrate layer (\textbf {a}) and output of ill-posedness (\textbf {b}). The data concerns the center of the domain ($y$=\SI {5}{m}).}}{38}{figure.22}
\contentsline {figure}{\numberline {23}{\ignorespaces Time spent in each module.}}{40}{figure.23}
\contentsline {figure}{\numberline {24}{\ignorespaces Initial bed elevation of the model of the experiments by \citet {Ashida90}.}}{41}{figure.24}
\contentsline {figure}{\numberline {25}{\ignorespaces Grain size distribution of the sediment in Case A2, original (blue) and discretized (orange).}}{42}{figure.25}
\contentsline {figure}{\numberline {26}{\ignorespaces Measured bed elevation of case A1 (from \citet {Ashida90}).}}{43}{figure.26}
\contentsline {figure}{\numberline {27}{\ignorespaces Predicted bed elevation of Case A1 without secondary flow (Simulation A1.1).}}{43}{figure.27}
\contentsline {figure}{\numberline {28}{\ignorespaces Predicted bed elevation of Case A1 with secondary flow and $D_H=\SI {1.6907e-04}{m^2/s}$ (Simulation A1.2).}}{44}{figure.28}
\contentsline {figure}{\numberline {29}{\ignorespaces Predicted bed elevation of Case A1 with secondary flow and $D_H=\SI {0.50}{m^2/s}$ (Simulation A1.3).}}{44}{figure.29}
\contentsline {figure}{\numberline {30}{\ignorespaces Longitudinal profile of bed elevation with time at $y=\SI {0.0466}{m}$ to the left of the centerline ($N=12$). The results correspond to the Simulation of Case A1 with secondary flow and $D_H=\SI {1.6907e-04}{m^2/s}$ (Simulation A1.2).}}{45}{figure.30}
\contentsline {figure}{\numberline {31}{\ignorespaces Secondary flow intensity of Case A1 with $D_H=\SI {1.6907e-04}{m^2/s}$ (Simulation A1.2).}}{46}{figure.31}
\contentsline {figure}{\numberline {32}{\ignorespaces Secondary flow intensity of Case A1 with $D_H=\SI {0.50}{m^2/s}$ (Simulation A1.3).}}{46}{figure.32}
\contentsline {figure}{\numberline {33}{\ignorespaces Longitudinal profile of bed elevation with time at $y=\SI {0.0466}{m}$ to the left of the centerline ($N=12$). The results correspond to the Simulation of Case A2 using the bed slope effects relation by \citet {Talmon95} (Simulation A2.1).}}{47}{figure.33}
\contentsline {figure}{\numberline {34}{\ignorespaces Longitudinal profile of bed elevation with time at $y=\SI {0.0466}{m}$ to the left of the centerline ($N=12$). The results correspond to the Simulation of Case A2 using the bed slope effects relation by \citet {Sekine92_1} (Simulation A2.2).}}{47}{figure.34}
\contentsline {figure}{\numberline {35}{\ignorespaces Measured mean grain size at the bed surface of case A2 (from \citet {Ashida90}).}}{48}{figure.35}
\contentsline {figure}{\numberline {36}{\ignorespaces Predicted mean grain size at the bed surface of case A2 (Simulation A2.3).}}{48}{figure.36}
\contentsline {figure}{\numberline {37}{\ignorespaces Longitudinal profile of bed elevation with time at $y=\SI {0.0466}{m}$ to the left of the centerline ($N=12$). The results correspond to the Simulation of Case A3.}}{49}{figure.37}
\contentsline {figure}{\numberline {38}{\ignorespaces Rhine at its entrance to The Netherlands. The red mesh is the computational grid (Bovenrijn). (Adapted from Google Earth)}}{50}{figure.38}
\contentsline {figure}{\numberline {39}{\ignorespaces Initial flow depth.}}{51}{figure.39}
\contentsline {figure}{\numberline {40}{\ignorespaces Initial mean grain size at the bed surface.}}{52}{figure.40}
\contentsline {figure}{\numberline {41}{\ignorespaces Initial mean grain size at the top substrate.}}{52}{figure.41}
\contentsline {figure}{\numberline {42}{\ignorespaces Ill-posed locations at the end of the simulation.}}{53}{figure.42}
\contentsline {figure}{\numberline {43}{\ignorespaces Locations that at some point have been ill-posed.}}{53}{figure.43}
\contentsline {figure}{\numberline {44}{\ignorespaces Bed elevation with time along a cross section in which the nodes are mainly well-posed (M=50; N=100:130).}}{54}{figure.44}
\contentsline {figure}{\numberline {45}{\ignorespaces Bed elevation with time along a cross section in which the nodes are mainly ill-posed (M=60; N=80:110).}}{54}{figure.45}
\contentsline {figure}{\numberline {46}{\ignorespaces Ill-posedness flow chart.}}{58}{figure.46}