Notation#
The consistent use of parameters and symbols for parameters is summarized in a notation table stating symbols, associated parameter definitions, and parameter units. Table 17 and Table 18 list Latin and Greek letters (symbols) used in this eBook.
Letter 
Unit 
Description 

\(A\) 
m\(^2\) 
flow cross section 
\(B\) 
m 
channel width at the water surface 
\(b\) 
m 
channel bottom width 
\(b_m\) 
m 
mean flow width 
\(c_{cfl}\) 
\(\) 
CourantFriedrichsLewy (CFL) condition 
\(c_{eq}\) 
g m\(^{3}\) 
fine material equilibrium nearbed concentration (cf. Suspended Load Formulae) 
\(C_{D}\) 
\(\) 
drag coefficient (cf. Equation (16)) 
\(c_{f}\) 
\(\) 

\(c'_{f}\) 
\(\) 
skin friction coefficient (Equation (12)) 
\(c_{mud}\) 
g m\(^{3}\) 
fine material concentration (cf. Equation (14)) 
\(c_{\varepsilon}\) 
\(\) 
convergence constant (cf. Equation (19)) 
\(cHSI\) 
index 
combined habitat suitability index 
\(D\) 
m\(^2\)/s 
diffusion coefficient (or diffusivity) 
\(D_{m}\) 
m 
mean grain diameter of a sediment mixture 
\(D_{pq}\) 
m 
grain diameter of which \(pq\)~\(\%\) of the mixture are finer 
\(D_{x}\) 
m 
dimensionless grain diameter (cf. Equation (6) and Shields parameter) 
\(Fr\) 
\(\) 

\(f_D\) 
\(\) 
DarcyWeisbach friction factor 
\(F_{eb}\) 
\(\) 
EinsteinBrown (EB) factor (Equation (5)) 
\(f_{eh}\) 
\(\) 
factor in the Engelund and Hansen bedload equation (8) 
\(f_{fr}\) 
\(\) 
friction correction factor for bed shear stress (Equation (11)) 
\(f_{k'_{s}}\) 
\(\) 

\(f_{mpm}\) 
\(\) 
MeyerPeter and Müller (MPM) factor (Equation (3)) 
\(g\) 
m s\(^{2}\) 
gravitational acceleration 
\(HSI\) 
index 
habitat suitability index 
\(h\) 
m 
water depth 
\(i\) or 
\(\) 
level one scalar iterator (1d space) 
\(j\) or 
\(\) 
level two scalar iterator (2d space) 

\(\) 
level three scalar iterator (3d space) 
\(k\) 
m\(^2\) s\(^{2}\) or J kg\(^{1}\) 

\(k_{st}\) 
m\(^{1/3}\) s\(^{1}\) 
Strickler roughness coefficient (fictive units) 
\(M\) 
kg m\(^{2}\) s\(^{1}\) 

\(m\) 
\(\) 
channel bank slope 
\(N\) 
\(\) 
target value of a matrix iterator 
\(n\) 
\(\) 
target value of a scalar iterator 
\(n_m\) 
m\(^{1/3}\) s 
Manning’s roughness coefficient (fictive units) 
\(P\) 
m 
wetted perimeter 
\(Pr\) 
\(\) 
probability 
\(Q\) (also \(Q_i\) or \(Q_j\)) 
m\(^3\) s\(^{1}\) 
discharge (water), fluxes, or volume flow rate 
\(q\) 
m\(^2\) s\(^{1}\) 
unit discharge 
\(Q_{b}\) 
kg s\(^{1}\) 
bed load transport (capacity) 
\(Q_{b * cr}\) 
kg s\(^{1}\) 
dimensionless bed load transport (capacity) 
\(q_{b}\) 
kg s\(^{1}\) m\(^{1}\) 
unit bedload transport (capacity) 
\(q_{b,sc}\) 
kg s\(^{1}\) m\(^{1}\) 
slopecorrected unit bedload transport (capacity) 
\(Q_{bf}\) 
m\(^3\) s\(^{1}\) 
bankfull discharge 
\(q_{s}\) 
kg s\(^{1}\) m\(^{1}\) 
unit sediment transport capacity 
\(q_{s,dep}\) 
kg s\(^{1}\) m\(^{1}\) 
unit suspended deposition flux (Equation (14)) 
\(q_{s,dep}\) 
kg s\(^{1}\) m\(^{1}\) 
unit suspended erosion flux (Equation (17)) 
\(Re\) 
\(\) 
Reynolds number 
\(R_h\) 
m 
hydraulic radius 
\(S\) 
\(\) 
slope 
\(S_0\) 
\(\) 
channel slope 
\(S_{e}\) 
\(\) 
energy slope 
\(s\) 
\(\) 
ratio of sediment grain and water density 
\(T\) 
years 
recurrence interval 
\(t\) 
s 
time, duration 
\(u\) or \(u_j\) or \(u_k\) 
m s\(^{1}\) 
flow velocity in \(x\), \(j\), and \(k\) directions, respectively 
\(\mathbf{u}\) (bold) 
m s\(^{1}\) 
flow velocity vector (multidimensional) 
\(u_{*}\) 
m s\(^{1}\) 
shear velocity 
\(u_{cr}\) 
m s\(^{1}\) 
critical shear velocity for mud deposition (cf. Equation (14)) 
\(w_{s}\) 
m s\(^{1}\) 
settling velocity (cf. Equation (15)) 
\(wse\) 
m a.s.l. 
water surface elevation (absolute) 
\(x\) 
m 
streamwise coordinate pointing in the upstream direction, or Easting of geodata 
\(y\) 
m 
spanwise coordinate pointing toward the right bank, or Northing of geodata 
\(z\) 
m 
vertical coordinate pointing against the gravity acceleration vector 
\(z_{b}\) 
m or m a.s.l. 
riverbed elevation pointing against the gravity acceleration vector 
Letter 
Unit 
Description 

\(\alpha\) 
rad or deg 
angle between the longitudinal channel (\(x\)) axis and a mass transport vector 
\(\beta\) 
\(\) 
empiric bedload intensity correction factor (e.g., in Gaia) 
\(\Delta {t}\) 
s or years 
time period (duration) or timestep length 
\(\Delta {x}\) 
m 
horizontal distance or cell size in \(x\)direction 
\(\Delta {y}\) 
m 
spanwise distance or cell size in \(y\)direction 
\(\Delta {z}\) 
m 
difference in height or cell size in \(z\)direction 
\(\epsilon\) 
\(\) 
porosity 
\(\varepsilon\) 
var. 
absolute error between two quantities (see Equation (18)) 
\(\eta\) 
m 
active layer thickness 
\(\eta_L\) 
\(\) 
Kolmogorov length scale 
\(\eta_T\) 
\(\) 
Kolmogorov time scale 
\(\eta_U\) 
\(\) 
Kolmogorov velocity scale 
\(\nabla\) 
\(\) 
operator vector (nabla) of partial differentials \(\frac{\partial}{\partial x_i}\) where \(x_i\) refers to the dimensions of the flow field [KC08] 
\(\mu\) = \(\nu \cdot \rho_w\) 
kg m\(^{1}\) s\(^{1}\) 
dynamic viscosity 
\(\nu\) = \(\mu \cdot \rho_w^{1}\) 
m\(^{2}\) s\(^{1}\) 
kinematic viscosity 
\(\Phi\) 
\(\) 
dimensionless Sediment transport 
\(\Phi_b\) 
\(\) 
dimensionless Bedload transport 
\(\psi\) 
variable 
constant of a transported particle (substance) 
\(\rho_s\) 
kg m\(^{3}\) 
sediment grain density 
\(\rho_w\) 
kg m\(^{3}\) 
density of water 
\(\tau\) 
N m\(^{2}\) 
bed shear stress 
\(\tau_{cr}\) 
N m\(^{2}\) 
Critical dimensional bed shear stress (cf. Equation (17)) 
\(\tau_x\) 
\(\) 

\(\tau_{x,cr}\) 
\(\) 
Critical Dimensionless bed shear stress or Shields parameter 