The Bedform Transport Equation
Flow conditions in the Delta of the Sacramento and San Joaquin Rivers are ideal for bedload measurements by monitoring bedform movement. Sand waves are tall and well formed, and the water surface is usually calm enough to use as a stable reference for transferring datum.
The general method of dune tracking has been practiced in the USGS at least since the 1950s. Special studies of bedform transport were made in the Rio Grande in the 1960s, where a simple equation for bedform transport was tested:
where Qw is the weight rate of sediment transport per unit width,
c is the average bedform celerity,
h is is the average bedform height,
and the specific weight of migrating sand is about 1900 kilograms per cubic meter.
The bedform height and celerity are derived from sonar profiles of the channel bed as measured from small boats with receivers for the Global Positioning System (GPS). Sonar transducers are mounted from the sides of the boat on vertical pipes, and a laptop computer records the GPS and sonar signals synchronously.
The Global Positioning System allows us to record our geographic location in the Delta within 1 meter on a second-by-second basis. We attach the antenna near the pipe that holds the sonar transducer on the boat and under the water. As the sonar records a depth every second, we know the geographic location of that depth.
Then, we assume that
(1) the water is level,
(2) we know the distance of the transducer under the water, and
(3) we know the water elevation from a nearby gage.
With some subtractions, the bed elevation at a precise geographic location at that moment is recorded.
With the measuring equipment in place, we follow a planned set of tracks on a map on the computer screen. As the boat moves, the GPS position is displayed on the map to show our progress along the planned tracks.
At the end of a track, only part of the equation for bedform movement has been derived. An initial position has been recorded for many bedforms at intervals spaced across the channel. One can derive the average bedform height from the profiles.
To derive the bedform celerity, the same profile is revisited after a sufficient time has elapsed for bedform migration to be accurately measurable.
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