Notation

$A$

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

$C$

g l$^{-1}$

depth-averaged suspended sediment concentration (cf. Equation (1))

$C_{eq}$

g l$^{-1}$

equilibrium near-bed (reference-level) suspended sediment concentration (cf. Suspended Load Formulae)

$C_{z_{ref}}$

g l$^{-1}$

near-bed suspended sediment concentration at the reference elevation $z_{ref}$

$C_{D}$

$-$

drag coefficient (cf. Settling velocity)

$C_{w1}$

$-$

Spalart-Allmaras closure constant in the destruction term ($\approx 0.3$)

$c_{f}$

$-$

combined form drag and skin friction coefficient

$c'_{f}$

$-$

skin friction coefficient (Equation (11))

$c_{mud}$

g l$^{-1}$

fine material (mud) concentration (cf. Equation (3))

$c_{\varepsilon}$

$-$

convergence constant (cf. Equation (2))

$cHSI$

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 (5) and Shields parameter)

$Fr$

$-$

Froude number

$f_D$

$-$

Darcy-Weisbach friction factor

$F_{eb}$

$-$

Einstein-Brown (EB) factor (Equation (4))

$f_{eh}$

$-$

factor in the Engelund and Hansen bedload equation (7)

$f_{fr}$

$-$

friction correction factor for bed shear stress (Equation (10))

$f_{k'_{s}}$

$-$

ratio between skin friction and mean diameter

$f_{mpm}$

$-$

Meyer-Peter and Müller (MPM) factor (Equation (2))

$f_w$

$-$

Spalart-Allmaras near-wall correction function in the destruction term

$g$

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

$k_s$

m

total bed roughness length (height)

$k'_{s}$

m

skin friction roughness length (cf. Equation (11))

$k_{sr}$

m

rippled-bed roughness length (cf. Suspended Load Formulae)

$k_{st}$

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

Strickler roughness coefficient (fictive units)

$l_w$

m

shortest distance from a mesh node to the nearest solid lateral boundary (wall distance); used in the Spalart-Allmaras destruction term

$M$

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

Partheniades (1965) erosion constant (cf. Equation (8))

$m$

$-$

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 (3))

$q_{s,dep}$

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

unit suspended erosion flux (Equation (8))

$Re$

$-$

Reynolds number

$Re_p$

$-$

particle Reynolds number ($Re_p = w_s D / \nu$, cf. Additional Sediment Parameters)

$R_h$

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 (3))

$w_{s}$

m s$^{-1}$

settling velocity (cf. Equation (4) and Settling velocity)

$w_{s,h}$

m s$^{-1}$

hindered settling velocity at high concentrations (cf. Additional Sediment Parameters)

$wse$

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

$z_{ref}$

m

reference (near-bed) elevation for suspended sediment exchange (cf. Suspended Load Formulae)

$\alpha$

rad or deg

angle between the longitudinal channel ($x$) axis and a mass transport vector

$\alpha_{k_s}$

$-$

ratio between skin friction roughness and mean grain diameter (RATIO BETWEEN SKIN FRICTION AND MEAN DIAMETER keyword; same quantity as $f_{k'_{s}}$)

$\beta$

$-$

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 (1))

$\varepsilon_s$

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

sediment turbulent diffusivity (cf. Equation (2))

$\eta$

m

active layer thickness

$\eta_L$

$-$

Kolmogorov length scale

$\eta_T$

$-$

Kolmogorov time scale

$\eta_U$

$-$

Kolmogorov velocity scale

$\theta'$

$-$

skin-friction–corrected Dimensionless bed shear stress ($\theta' = \mu\theta$), i.e., the Shields parameter-scale shear used in the suspension formulae; equivalent to $\tau_{x}$ with skin friction correction (cf. Suspended Load Formulae)

$\theta_{cr}$

$-$

critical Shields parameter (equivalent to $\tau_{x,cr}$)

$\nabla$

$-$

operator vector (nabla) of partial differentials $\frac{\partial}{\partial x_i}$ where $x_i$ refers to the dimensions of the flow field Kundu & Cohen, 2008

$\mu$ = $\nu \cdot \rho_w$

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

dynamic viscosity

$\nu$ = $\mu \cdot \rho_w^{-1}$

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

kinematic viscosity

$\nu_t$

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

eddy (turbulent) viscosity

$\tilde{\nu}$

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

modified turbulent kinematic viscosity; transport variable in the Spalart-Allmaras turbulence model

$\Phi$

$-$

dimensionless Sediment transport

$\Phi_b$

$-$

dimensionless Bedload transport

$\phi$

$-$

volumetric sediment concentration (cf. hindered settling, Additional Sediment Parameters)

$\psi$

variable

constant of a transported particle (substance)

$\rho_s$

kg m$^{-3}$

sediment grain density

$\rho_w$

kg m$^{-3}$

density of water

$\sigma_s$

$-$

Schmidt number relating sediment diffusivity to eddy viscosity (cf. Equation (2); Gaia fixes $\sigma_s = 1.0$)

$\tau$

N m$^{-2}$

bed shear stress

$\tau_{cr}$

N m$^{-2}$

Critical dimensional bed shear stress (cf. Equation (8))

$\tau_x$

$-$

Dimensionless bed shear stress

$\tau_{x,cr}$

$-$

Critical Dimensionless bed shear stress or Shields parameter