Stream power

Stream Power
Water flowing in creek looped As water flows through the river, it drags along the bottom and sides exerting a force on them that this force is called stream power and can cause rocks to move or material from the banks to dislodge resulting in erosion.
Common symbols	Ω, ω
SI unit	Watts
In SI base units	kg m2 s−3
Derivations from other quantities	Ω=ρgQS
Dimension	M L2 T−3

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. Stream power is a valuable measurement for hydrologists and geomorphologist tackling sediment transport issues as well as for civil engineers using it in the planning and construction of roads, bridges, 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 quantifying the same thing, this lead 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:

${\frac {\Delta PE}{\Delta t}}=mg{\frac {\Delta z}{\Delta t}}$

where water mass and gravitational acceleration are constant. We can use the channel slope and the stream velocity as a stand-in for ${\Delta z}/{\Delta t}$: the water will lose elevation at a rate given by the downward component of velocity $u_{z}$. For a channel slope (as measured from the horizontal) of $\alpha$:

${\frac {\Delta z}{\Delta t}}=u_{z}=u\sin(\alpha )\approx uS$

where $u$ is the downstream flow velocity. It is noted that for small angles, $\sin(\alpha )\approx \tan(\alpha )=S$. Rewriting the first equation, we now have:

${\frac {\Delta PE}{\Delta t}}=mguS$

Remembering that power is energy per time and using the equivalence between work against the bed and loss in potential energy, we can write:

$\Omega ={\frac {\Delta PE}{\Delta t}}$

Finally, we know that mass is equal to density times volume. From this, we can rewrite the mass on the right hand side

$m=\rho Lbh$

where $L$ is the channel length, $b$ is the channel width (breadth), and $h$ is the channel depth (height). We use the definition of discharge

$Q=ubh$

where $A$ 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.

$\Omega =\rho gQ{\cancelto {1}{L}}S$

Unit stream power is stream power per unit channel width, and is given by the equation:

$\omega ={\frac {\rho gQS}{b}}$

where ω is the unit stream power, and b is the width of the channel.

Stream power is used extensively in models of landscape evolution and river incision. Unit stream power is often used for this, because simple models use and evolve a 1-dimensional downstream profile of the river channel. It is also used with relation to river channel migration, and in some cases is applied to sediment transport.