Stream power

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.

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. We know that change in potential energy over change in time is given by the equation: \frac{\Delta PE}{\Delta t} = m g \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 u S 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} = m g u S 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 L b h 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 = u b h 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 g Q \cancelto{1}{L} S 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: \Omega=\rho g Q S where Ω is the stream power, ρ is the density of water (1000 kg/m 3 ), g is acceleration due to gravity (9.8 m/s 2 ), Q is discharge (m 3 /s), and S is the channel slope. 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: Total\ stream \ power = \Omega\ L where Ω is the stream power, per unit downstream length and L is the length of the stream. Unit stream power is stream power per unit channel width, and is given by the equation: \omega=\frac{\rho g Q S}{b} where ω is the unit stream power, and b is the width of the channel. Normalizing the stream power by the width of the river allows for a better comparison between rivers of various widths. This also provides a better estimation of the sediment carrying capacity of the river as wide rivers with high stream power are exerting less force per surface area than a narrow river with the same stream power, as they are losing the same amount of energy but in the narrow river it is concentrated into a smaller area. Critical unit stream power is the amount of stream power needed to displace a grain of a specific size, it is given by the equation: \omega_0= \tau_0\nu_0 where τ 0 is the critical shear stress of the grain size that will be moved which can be found in the literature or experimentally determined while v 0 is the critical mobilization speed. Critical stream power can be used to determine the stream competency of a river, which is a measure to determine the largest grain size that will be moved by a river. In rivers with large sediment sizes the relationship between critical unit stream power and sediment diameter displaced can be reduced to: \omega_0=0.030D_i^{1.69} While in intermediate-sized rivers the relationship was found to follow: \omega_0=0.130D_i^{1.438} Shear stress is another variable used in erosion and sediment transport models representing the force applied on a surface by a perpendicular force, and can be calculated using the following formula \tau=hS \rho g Where is the shear stress, S is the slope of the water, ρ is the density of water (1000 kg/m 3 ), g is acceleration due to gravity (9.8 m/s 2 ). Shear stress can be used to compute the unit stream power using the formula \omega = \tau \ V Where V is the velocity of the water in the stream. Stream power has also been used as a criterion to determine whether a river is in a state of reshaping itself or whether it is stable. A value of unit stream power between 30 and 35 W m −2 in which this transition occurs has been found by multiple studies. Another technique gaining popularity is using a gradient of stream power by comparing the unit stream power upstream to the local unit stream power ( \Delta\omega=\omega_{local}-\omega_{upstream} ) to identify patterns such as sudden jumps or drops in stream power, these features can help identify locations where the local terrain controls the flow or widens out as well as areas prone to erosion.

Text derived from https://en.wikipedia.org/w/index.php?oldid=1353331335 under CC BY-SA 4.0; modifications have been made.