# 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.

Table 17 Latin letters (symbols) and parameters used in this eBook (in alphabetical order).#

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))

$$c_{f}$$

$$-$$

combined form drag and skin friction coefficient

$$c'_{f}$$

$$-$$

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)

$$Fr$$

$$-$$

Froude number

$$f_D$$

$$-$$

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))

$$f_{k'_{s}}$$

$$-$$

ratio between skin friction and mean diameter

$$f_{mpm}$$

$$-$$

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 i

$$-$$

level one scalar iterator (1d space)

$$j$$ or j

$$-$$

level two scalar iterator (2d space)

k

$$-$$

level three scalar iterator (3d space)

$$k$$

m$$^2$$ s$$^{-2}$$ or J kg$$^{-1}$$

turbulent kinetic energy

$$k_{st}$$

m$$^{1/3}$$ s$$^{-1}$$

Strickler roughness coefficient (fictive units)

$$M$$

kg m$$^{-2}$$ s$$^{-1}$$

Partheniades [Par65] erosion constant (cf. Equation (17))

$$m$$

$$-$$

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$$^{-1}$$

$$Q_{b * cr}$$

kg s$$^{-1}$$

$$q_{b}$$

kg s$$^{-1}$$ m$$^{-1}$$

$$q_{b,sc}$$

kg s$$^{-1}$$ m$$^{-1}$$

$$Q_{bf}$$

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

$$S$$

$$-$$

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

Table 18 Greek letters (symbols) and parameters used in this eBook (in alphabetical order).#

Letter

Unit

Description

$$\alpha$$

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

$$\Phi_b$$

$$-$$

$$\psi$$

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))

$$\tau_x$$

$$-$$

Dimensionless bed shear stress

$$\tau_{x,cr}$$

$$-$$