Stream Power

Stream power is the rate of potential energy loss per unit of channel length.^[7] This potential energy is lost moving particles along the stream bed.

$${\displaystyle \Omega =\rho _{w}gQS}$$

where ${\displaystyle \Omega }$ is the stream power, ${\displaystyle \rho _{w}}$ is the density of water, ${\displaystyle g}$ is the gravitational acceleration, ${\displaystyle S}$ is the channel slope, and ${\displaystyle Q}$ is the discharge of the stream.

The discharge of a stream, ${\displaystyle Q}$, is the velocity of the stream, ${\displaystyle U}$, multiplied by the cross-sectional area, ${\displaystyle A_{\mathrm {CS} }}$, of the stream channel at that point:

$${\displaystyle Q=UA_{\mathrm {CS} }}$$

in which ${\displaystyle Q}$ is the discharge of the stream, ${\displaystyle U}$ is the average stream velocity, and ${\displaystyle A_{\mathrm {CS} }}$ is the cross-sectional area of the stream.

As velocity increases, so does stream power, and a larger stream power corresponds to an increased ability to move bed load particles.

Shear Stress and Critical Shear Stress

In order for sediment transport to occur in gravel bed channels, flow strength must exceed a critical threshold, called the critical threshold of entrainment, or threshold of mobility. Flow over the surface of a channel and floodplain creates a boundary shear stress field. As discharge increases, shear stress increases above a threshold and starts the process of sediment transport. A comparison of the flow strength available during a given discharge to the critical shear strength needed to mobilize the sediment on the bed of the channel helps us predict whether or not sediment transport is likely to occur, and to some degree, the sediment size likely to move. Although sediment transport in natural rivers varies wildly, relatively simple approximations based on simple flume experiments are commonly used to predict transport.^[8] Another way to estimate stream competency is to use the following equation for critical shear stress, ${\displaystyle \tau _{c}}$ which is the amount of shear stress required to move a particle of a certain diameter.^[9]

$${\displaystyle \tau _{c}=\tau _{c}^{\ast }(\rho _{s}-\rho _{w})gd_{50}}$$

where:

${\displaystyle \tau _{c}^{\ast }=}$ Shields parameter, a dimensionless value which describes the resistance of the stream bed to gravitational acceleration, also described as roughness or friction,

${\displaystyle \rho _{s}=}$ Particle density, and ${\displaystyle \rho _{s}-\rho _{w}}$ is the effective density of the particle when submerged in water (Archimedes principle).^[10]

${\displaystyle g=}$ Gravitational acceleration.

${\displaystyle d_{50}=}$ grain diameter, usually measured as d50 which is the median particle diameter when sampling particle diameters in a stream transect.

The shear stress of a stream is represented by the following equation:

$${\displaystyle \tau =\rho _{w}gDS}$$

where:

${\displaystyle D=}$ average depth

${\displaystyle S=}$ stream slope.

If we combine the two equations we get:

$${\displaystyle \rho _{w}gDS=\tau _{c}^{\ast }(\rho _{s}-\rho _{w})gd_{50}}$$

Solving for particle diameter d we get

$${\displaystyle d_{50}={\frac {\rho _{w}DS}{\tau _{c}^{\ast }(\rho _{s}-\rho _{w})}}}$$

The equation shows particle diameter, ${\displaystyle d_{50}}$, is directly proportional to both the depth of water and slope of stream bed (flow and velocity), and inversely proportional to Shield's parameter and the effective density of the particle.

Lift

Velocity differences between the bottom and tops of particles can lead to lift. Water is allowed to flow above the particle but not below resulting in a zero and non-zero velocity at the bottom and top of the particle respectively. The difference in velocities results in a pressure gradient that imparts a lifting force on the particle. If this force is greater than the particle's weight, it will begin transport.^[11]

Turbulence

Flows are characterized as either laminar or turbulent. Low-velocity and high-viscosity fluids are associated with laminar flow, while high-velocity and low-viscosity are associated with turbulent flows. Turbulent flows result velocities that vary in both magnitude and direction. These erratic flows help keep particles suspended for longer periods of time. Most natural channels are considered to have turbulent flow.^[7]

Cohesion

Another important property comes into play when discussing stream competency, and that is the intrinsic quality of the material. In 1935 Filip Hjulström published his curve, which takes into account the cohesiveness of clay and some silt. This diagram illustrates stream competency as a function of velocity.^[12]

By observing the size of boulders, rocks, pebbles, sand, silt, and clay in and around streams, one can understand the forces at work shaping the landscape. Ultimately these forces are determined by the amount of precipitation, the drainage density, relief ratio and sediment parent material.^[7] They shape depth and slope of the stream, velocity and discharge, channel and floodplain, and determine the amount and kind of sediment observed. This is how the power of water moves and shapes the landscape through erosion, transport, and deposition, and it can be understood by observing stream competency.

Bedrock

Stream competence does not rely solely on velocity. The bedrock of the stream influences the stream competence. Differences in bedrock will affect the general slope and particle sizes in the channel. Stream beds that have sandstone bedrock tend to have steeper slopes and larger bed material, while shale and limestone stream beds tend to be shallower with smaller grain size.^[6] Slight variations in underlying material will affect erosion rates, cohesion, and soil composition.

Vegetation^[13]

Vegetation has a known impact on a stream's flow, but its influence is hard to isolate. A disruption in flow will result in lower velocities, leading to a lower stream competence. Vegetation has a 4-fold effect on stream flow: resistance to flow, bank strength, nucleus for bar sedimentation, and construction and breaching of log-jams.

Resistance to flow

Cowan method for estimating Manning's n.

$${\displaystyle n=(n_{0}+n_{1}+n_{2}+n_{3}+n_{4})m_{5}}$$

Manning's n considers a vegetation correction factor. Even stream beds with minimal vegetation will have flow resistance.