Stream power, originally derived by R. A. Bagnold in the 1960s, is the amount of energy the water in a river or stream is exerting on the sides and bottom of the river. Stream power is the result of multiplying the density of the water, the acceleration of the water due to gravity, the volume of water flowing through the river, and the slope of that water. There are many forms of the stream power formula with varying utilities, such as comparing rivers of various widths or quantifying the energy required to move sediment of a certain size. Stream power is closely related to other criteria such as stream competency and shear stress. Stream power is a valuable measurement for hydrologists and geomorphologists tackling sediment transport issues as well as for civil engineers, who use it in the planning and construction of roads, bridges, dams, and culverts.
History
Although many authors had suggested the use of power formulas in sediment transport in the decades preceding Bagnold's work, and in fact Bagnold himself suggested it a decade before putting it into practice in one of his other works, it wasn't until 1966 that R. A. Bagnold tested this theory experimentally to validate whether it would indeed work or not. This was successful and since then, many variations and applications of stream power have surfaced. The lack of fixed guidelines on how to define stream power in this early stage lead to many authors publishing work under the name "stream power" while not always measuring the entity in the same way; this led to partially failed efforts to establish naming conventions for the various forms of the formula by Rhoads two decades later in 1986. Today stream power is still used and new ways of applying it are still being discovered and researched, with a large integration into modern numerical models utilizing computer simulations.
Derivation
It can be derived by the fact that if the water is not accelerating and the river cross-section stays constant (generally good assumptions for an averaged reach of a stream over a modest distance), all of the potential energy lost as the water flows downstream must be used up in friction or work against the bed: none can be added to kinetic energy. Therefore, the potential energy drop is equal to the work done to the bed and banks, which is the stream power.
We know that change in potential energy over change in time is given by the equation:where water mass and gravitational acceleration are constant. We can use the channel slope and the stream velocity as a stand-in for : the water will lose elevation at a rate given by the downward component of velocity . For a channel slope (as measured from the horizontal) of :where is the downstream flow velocity. It is noted that for small angles, . Rewriting the first equation, we now have:Remembering that power is energy per time and using the equivalence between work against the bed and loss in potential energy, we can write:Finally, we know that mass is equal to density times volume. From this, we can rewrite the mass on the right hand sidewhere is the channel length, is the channel width (breadth), and is the channel depth (height). We use the definition of dischargewhere is the cross-sectional area, which can often be reasonably approximated as a rectangle with the characteristic width and depth. This absorbs velocity, width, and depth. We define stream power per unit channel length, so that term goes to 1, and the derivation is complete.
Various Forms
(Total) Stream power
Stream power is the rate of energy dissipation against the bed and banks of a river or stream per unit downstream length. It is given by the equation:where Ω is the stream power, ρ is the density of water (1000 kg/m3), g is acceleration due to gravity (9.8 m/s2), Q is discharge (m3/s), and S is the channel slope.
Total Stream Power
Total stream power often refers simply to stream power, but some authors use it as the rate of energy dissipation against the bed and banks of a river or stream per entire stream length. It is given by the equation:where Ω is the stream power, per unit downstream length and L is the length of the stream.