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 |
\(c_{cfl}\) |
\(-\) |
Courant-Friedrichs-Lewy (CFL) condition |
\(c_{eq}\) |
g m\(^{-3}\) |
fine material equilibrium near-bed 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) |
\(Fr\) |
\(-\) |
|
\(f_D\) |
\(-\) |
Darcy-Weisbach friction factor |
\(F_{eb}\) |
\(-\) |
Einstein-Brown (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)) |
\(f_{k'_{s}}\) |
\(-\) |
|
\(f_{mpm}\) |
\(-\) |
Meyer-Peter 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\(^{-1}\) |
slope-corrected unit bedload transport (capacity) |
\(Q_{bf}\) |
m\(^3\) s\(^{-1}\) |
bank-full 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 |