Ackers and White

The original Ackers and White (1973) was developed for to estimate the total load of uniform material. Ackers (1993) subsequently provided an update to the formula empirical coefficients. Day (1980) and Proffitt and Sutherland (1983) extended the original Ackers and White (1973) by multiplying it by a hiding and exposure correction factor. The fractional total-load transport potential is given by

1)	$\begin{array}{l}\displaystyle {q_{tk}}^* = ρ_wghU{X_{tk}}^*\end{array}$

where

$\begin{array}{l}{q_{tk}}^*\end{array}$ = sediment transport potential [M/L/T]

$\begin{array}{l}ρ_{w}\end{array}$ = water density [M/L³]

$\begin{array}{l}g\end{array}$ = gravity acceleration [L/T²]

$\begin{array}{l}h\end{array}$ = water depth [L]

$\begin{array}{l}U\end{array}$ = depth-averaged current velocity [L/T]

$\begin{array}{l}{X_{tk}}^*\end{array}$ = sediment concentration potential by weight

The sediment concentration potential is determined from

2)	$\begin{array}{l}\displaystyle \frac{X_{tk}^{}h\rho _{w}}{d_{k}\rho _{sk}}\left(\frac{u_{}}{U}\right)^{n}=\Lambda \left(\frac{F_{grk}}{A_{c}}-1\right)^{m}\end{array}$

where

$\begin{array}{l}F_{grk}\end{array}$ = sediment mobility factor [-]

$\begin{array}{l}u_{*}=\sqrt{\tau _{b}/\rho _{w}}\end{array}$ = bed shear velocity [M/T]

$\begin{array}{l}ρ_{sk}\end{array}$ = sediment density [M/L³]

$\begin{array}{l}\Lambda\end{array}$ = $\begin{array}{l}\Lambda (d_{*k})\end{array}$ = empirical coefficient [-]

$\begin{array}{l}A_{c}\end{array}$ = $\begin{array}{l}A_c(d_{*k})\end{array}$ = empirical coefficient [-]

$\begin{array}{l}n\end{array}$ = $\begin{array}{l}n(d_{* k})\end{array}$ = empirical exponent [-]

$\begin{array}{l}m\end{array}$ = $\begin{array}{l}m(d_{* k})\end{array}$ = empirical exponent [-]

The sediment mobility factor is given by

3)	$\begin{array}{l}\displaystyle F_{grk}=\eta _{k}\frac{u_{*}^{n}}{\sqrt{R_{k}gd_{k}}}\left[\frac{U}{\sqrt{32}\log _{10}(10h/d_{k})}\right]^{1-n}\end{array}$

where

$\begin{array}{l}R_{k}\end{array}$ = $\begin{array}{l}ρ_{sk}/ρ_w− 1\end{array}$ = submerged specific gravity of a particle [-]

$\begin{array}{l}ρ_{sk}\end{array}$ = sediment density [M/L³]

$\begin{array}{l}ρ_{w}\end{array}$ = water density [M/L³]

$\begin{array}{l}η_{k}\end{array}$ = hiding and exposure correction factor [-]

$\begin{array}{l}τ_{b}\end{array}$ = bed shear stress [M/L/T²]

$\begin{array}{l}d_{k}\end{array}$ = sediment diameter [L]

The firs term in the above equation corresponds to the bed-load transport while the second the suspended-load transport. The formula is based on the understanding that bed load is attributed to the grain shear stress while the suspended load is related to the turbulence intensity. The empirical coefficients were revised by Ackers (1993) as

4)

$\begin{array}{l}n=\begin{cases} 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,d_{*k}\leq 1\\ 1-0.56\log _{10}(d_{*k})\,\,\mathrm{for}\,\,1< d_{*k}\leq 60\\ 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,d_{*k}>60 \end{cases}\right.\end{array}$

5)	$\begin{array}{l}A_{c}=\begin{cases} 0.23d_{k}^{-1/2}+0.14\,\,\,\,\mathrm{for}\,\,d_{k}\leq 60\\ 0.17\,\,\text{otherwise} \end{cases}\right.\end{array}$

6)	$\begin{array}{l}m=\begin{cases} 6.83d_{k}^{-1}+1.67\,\,\,\,\mathrm{for}\,d_{k}\leq 60\\ 1.78\,\,\text{otherwise} \end{cases}\right.\end{array}$

7)	$\begin{array}{l}\Lambda =\left\{\begin{array} \, \exp \left\{ 2.791\log _{10}(d_{k})-0.98 \left[ \log _{10}(d_{k})\right]^{2}-3.46\right\}\,\,\,\,\,\mathrm{for}\,\,d_{*k}\leq 60\\ 0.0025\,\,\,\,\text{otherwise} \end{array}\right.\end{array}$

Wu (2007) tested and compared the Ackers-White (AW) formula for graded sediments and found that it tends to over predict the transport significantly for fine sediments less than 0.2 mm. However, the Ackers-White formula performs very well for uniform sediments. Day (1980) and Proffitt and Sutherland (1983) developed hiding and exposure correction factors for the Ackers-White formula and are available here to the user. It is noted that in AW, hiding and exposure is considered through a transport multiplication factor rather than through the sediment mobility.