Parker (1990)

The Parker (1990) bed-load formula¹ is only appropriate for calculation of bed-load in gravel bed streams.

The non-dimensional bed-load is defined by:

\[ W_{i}^{*} = \frac{\left( s_{i} - 1 \right)gq_{b,i}}{F_{i}\ \left( u_{f}' \right)^{3}} \]

Equation: Definition of the non-dimensional bed-load for a sediment fraction.

In the Parker formula the weight function \(F_{i}\) is renormalized to only include the gravel fractions.

The non-dimensional bed-load is calculated from the formula:

\[ W_{i}^{*} = 0.00218\ G(\phi_{i}) \]

Equation: Parker non-dimensional bed-load.

\(\phi_{i}\) is calculated from:

\[ \phi_{i} = \omega\ \phi_{0}\ \left( \frac{d_{i}}{d_{g}} \right)^{-0.0951} \]

Equation: Parameter φᵢ in the Parker bed-load formula.

Where \(d_{g}\) is the geometric mean grain size, and \(\phi_{0}\) is calculated from:

\[ \phi_{0} = \frac{\theta_{g}'}{\theta_{r}'} \]

Equation: Ratio of the geometric mean Shields parameter to the reference Shields parameter.

Where the skin friction Shields parameter \(\theta_{g}'\) is based on the geometric mean diameter:

\[ \theta_{g}' = \frac{\left( u_{f}' \right)^{2}}{(s - 1)gd_{g}} \]

Equation: Skin friction Shields parameter for the geometric mean diameter.

And the reference Shields parameter:

\[ \theta_{r}' = 0.0386 \]

Equation: Reference Shields parameter used in the Parker formula.

The G-function is calculated from:

\[ G(\phi) = \begin{cases} 5474\ \left( 1 - \dfrac{0.853}{\phi} \right)^{4.5}, & \phi > 1.59 \\ \exp\left( 14.2(\phi - 1) - 9.28(\phi - 1)^{2} \right), & 1 \leq \phi \leq 1.59 \\ \phi^{14.2}, & \phi < 1 \end{cases} \]

Equation: The G-function in the Parker bed-load formula.

Figure: Parker (1990) σ₀ and ω₀ functions.

The ω-function is given by:

\[ \omega = 1 + \frac{\sigma_{s}}{\sigma_{0}(\phi_{0})}\ \left( \omega_{0}\left( \phi_{0} \right) - 1 \right)\theta_{r}' = 0.0386 \]

Equation: The ω-function in the Parker bed-load formula.

Where the two functions \(\sigma_{0}\) and \(\omega_{0}\) are shown in the σ₀ and ω₀ chart.

Notes regarding Parker

Parker is only selectable for bed-load. The Parker formula was developed for multi-fraction bed-load calculations. The \(F_{i}\) weights are calculated separately and normalized for the gravel fractions.

Parker can be used also for single fraction bed-load.

Egiazaroff does not apply to Parker; it uses its own hiding function.

The GUI critical Shields parameter does not apply.

---

1. Parker, G. (1990). Surface-based bedload transport relation for gravel rivers, Journal of Hydraulic Research, 28(4): 417-436. ↩