# Stream Power Notes

Stream power is the rate the energy of flowing water is expended on the bed and banks of a channel. Said another way, it is the potential for flowing water to perform geomorphic work. Stream power can be calculated on the cheap from DEM data because of the area-discharge relationship, but recognize that the DEM doesn’t provide everything you need.

Here’s a quick, readable summary of Stream Power by Amanda Brown. Thanks, Amanda!
Stream Power Overview_Amanda Brown_May 2015.doc

Erosion Equation made by Megan Moeller

Stream power (watts, W) is not directly measured by sticking an instrument in a stream, rather, we use incomplete data and certain approximations/assumptions in calculating this index. Its important to recognize that there are several flavors of stream power (see Definitions below). Often, relative stream power values are calculated; true values are often less important than the pattern they form along a single channel profile or across a landscape (Finlayson & Montgomery, 2003; Golden & Springer, 2006; Perez-Pena et al., 2009). No one publishes meausured stream power values unless they own a large flume.

High stream power values generally correspond with steep, straight, scoured reaches, and bedrock gorges. Low stream power values occur in broad alluvial flats, floodplains, and slowly subsiding areas, where the valley fill is usually intact and deepening. Increasing the channel slope or discharge raises stream power values, thus the ability of channel to incise its bed. Greater sediment loads will deter incision to some extent, but the amount and calibre of alluvial cover will drive changes in the profile and river form (Seidl et al., 1994; Sklar & Dietrich, 2001; Finnegan et al., 2007).

Stream power laws describe a channel’s ability to move sediment, thus its potential to incise, widen, or aggrade. This has implications for flood hazard assessment. Incision is governed by several factors and the suite of equations reflect this. All the usually suspects are considered: conservation of water mass, gravity, basin hydrology, hydraulic geometry (discharge Q, channel slope S, channel width W), shear stresses, climate, concavity, flood interval, fracture spacing in bedrock, and bedrock erodibility are considered. Other factors such as grainsize of bedload material (Brummer & Montgomery, 2003; Wohl, 2000; Whipple, 2004), block quarrying (Dubinski & Wohl, 2013), and clast-scale processes like scour, plucking, abrasion (Tinkler & Wohl, 1998) can also be accounted for.

The spatial distribution of stream power along a channel has been linked to things like river form and flood-response behavior in many studies (Bagnold, 1966; 1980; Graf, 1983; Magilligan, 1992; Lecce, 1997; Knighton, 1999; Flores et al., 2006). While theory predicts stream power values along a channel long profile should peak in mid-basin reaches (Knighton, 1999), local variations in channel slope (and tributary effects on discharge) can be significant. Slope exerts an especially strong control on stream power and since many factors can affect the local slope of a channel, stream power can vary substantially at the reach scale (Fonstad, 2003; Reinfelds et al., 2003; Jain et al., 2006). When it comes to stream power along a channel, variability from one reach to another is probably the rule (Hack, 1973).

Discussions of climate, tectonics, river incision, channel stability, and landscape evolution nearly always involve the concept of stream power (Bagnold, 1966). Useful GIS workflow models are provided by Phillips et al. (2013, Fig. 3), Jain et al., (2006, Fig.2 ). Textbooks by Anderson & Anderson (2010, Chapter 13), Burbank & Anderson (2012, Chapter 11), and Fryirs & Brierley (2012) do an OK job of explaining stream power. The Wikipedia article on Sediment Transport is also good. See the equations below and the references at the end.

I’m still learning this stuff myself; we didn’t cover any of this as a geology undergrad. My graduate school instructors were pretty lousy teachers when it came to anything involving flowing water. I should have taken more classes from Harper and Humphreys. Today, I read the original journal articles, distill the information here, and check the authors’ dissertations when I need to know more about their equations and assumptions. I’ve never come across my former instructors’ dissertations. Weird.

DEFINITIONS

a.) Specific Stream Power (Ω) or Unit Stream Power (ω) or Mean Stream Power (ω) is power per unit area of a channel in watts per meter (W/m).

ω = ρ g R Se V  or  Ω / width of active channel

b.) Total Stream Power (ω) (Phillips, 1989) or Cross-sectional Stream Power (Ω) (Bagnold, 1966) is the power per unit length in watts per square meter (W/m2). Can also be expressed per unit bed area if you divide by channel width (ω = γ Q Se / w).

Ωρ g Q Seγ Q Se

c.) Gross Stream Power may be found when discharge for 5-, 10-, 50-, 100-year flood events is combined with slope (Reinfelds et al., 2004; Brierley & Fryirs, 2005).

d.) A less commonly used metric, Total Stream Power (P), may be calculated for an entire reach (Fonstad, 2003). Not the same as Total Stream Power (ω) described above.

e.) Critical Stream Power (ωc*) is a dimensionless term related to a grainsize (Di) that marks the incipient motion of particles on bed (Petit et al., 2005; Eaton & Church, 2011; Parker et al., 2011; Camenen et al., 2012).

ωc* = ωc / ρs(gRDi)3/2

EQUATIONS

Ω = ρ g H W U S      or      Ω = γ Q Se

Energy is lost by a flowing stream as it performs work on its rock bed, bed-covering sediments, and banks. Stream power indices are expressed in many ways (Rhodes, 1987; Brummer & Montgomery, 2003; Annandale, 2006). The literature is rife with inconsistencies in both stream power terms (specific, unit, mean, total, etc.) and symbols in stream power equations for (Ω, ω, P). Just know that when you see Ω and ω, you’re talking stream power index values.

Ω = ρ g Q S
Simplification of the Stream Power equation is possible since H x W x U = Q.

Ω = ρ g Qm S
It is also possible to calculate Mean Annual Stream Power. Qm in this case is mean annual discharge. While substituted for in other equations, Qm can be approximated at the region scale using a DEM and satellite data from a.) Moderate Resolution Imaging Spectroradiometer (MODIS), b.) Advanced Microwave Scanning Radiometer–Earth Observing System (AMSR-E), and c.) Tropical Rainfall Measuring Mission (TRMM) if appropriate precautions are taken and corrections made (Khan et al., 2012 in press; DeGroeve et al., 2010; Shaban et al., 2009; Finnegan et al., 2008; Brakenridge et al., 2005; Anders et al., 2006; Dartmouth Flood Observatory website).

ω = ρ g Q S / W

Unit Stream Power (ω) is found by dividing by the width of the channel (W).

ω = τb U
Shear stress at the channel bed is an equivalent expression to Unit Stream Power (ω) and channel width changes slowly with distance downstream.

kω = dz/dt = k ρ g Q S / W
Substitution allows the rewriting the Stream Power equation. dz/dt is the vertical change in the elevation of the channel bed over time (advection or erosion and removal of bed material). The coefficient k accounts for erodibility of the channel bed influenced by rock strength, fracture density, and fissibility (and perhaps more).

dz/dt = -δ / T
Expression for the mean rate of vertical lowering of a channel’s bed.

dz/dt = -(α δ / B E) ω
The mean lowering rate of the channel bed averaged over many detachments.

T = B E / α ω
The equation for time (T).

E ≈ A S
Erosion is proportional to product of Drainage Area (A) and Slope (S) according to Seidl & Dietrich (1992).

Q S ≈ A S
If precipitation is not highly non-uniform, then local discharge (Q) is proportional to Drainage Area (A). Find A at any point along a channel using the flow accumulation raster.

E = Keb – τc)
E is erosion rate; Fluvial incision into bedrock involves bed shear stress and a garbage can coefficient.

S = ks A
Slope of channel (S) varies as an inverse power law to drainage area (A).

E = K Am Sn
The stream power erosion law (Howard, 1994; Whipple & Tucker, 1999); E is erosion rate; m and n are positive values determined by the channel erosion process (Snyder et al., 2000; Hack, 1957; Seidl & Dietrich, 1992; Whipple & Tucker, 1999; Tucker & Whipple, 2002); A is area, a proxy for discharge (Wobus et al., 2006)

E = k1 Am Sn
A generalized form of the Stream Power equation, where k1 includes ρ, g, effective precipitation relating to A and Q of watershed, the scaling of W, and erodibility of bedrock.

dz/dt = 0     so     U – E = 0     so     U – K Am Sn = 0
U – K Am Sn is the change to a channel profile at a point through time. If Uplift (+z) = Erosion (-z), where z is vertical change, and erosion is uniform along the channel, then the channel remains at a constant elevation through time though material is being removed by the channel. When z is in equilibrium you have achieved steady state. This is the steady state equation, a starting point for discussion of bedrock erosion by channels.

dz/dt = U – K(P A)m Sn = 0
Because discharge (thus A) is a function of precipitation (P), “climate” in the form of precip can be introduced into the equation (D’Arcy & Whittaker, 2013; Whittaker et al., 2008; Whittaker & Boulton, 2012).

S = (U / Pm K)1/n A-m/n
Rearranging the precipitation-influenced equation to find local slope (S); this assumes steady state balance between uplift and erosion (dz/dt = 0; channel elevation does not change over time).

ks = (U / Pm K)1/n
Channel steepness index with precipitation, where the exponent m/n (often simplified to the concavity index, θ); the constant of proportionality between local slope and drainage area is termed ks, the channel steepness index (D’Arcy & Whittaker, 2013). Channel steepness is sensitive to uplift rate and precipitation (Wobus et al., 2006; Kirby & Whipple, 2012). ks is found by plotting in log-log space local slope (slope of channel pixels) vs. their upstream drainage area (flow accumulation of channel pixels).

S = (U / K)1/n A-m/n
Channel slope (S) is a function of drainage area (A) without precipitation factored in.

A = ka xh
Area (A) may be replaced by downstream distance from the watershed divide using Hack’s Law (Hack, 1957).

m/n = log(St / Sp) / log(Ap / At)
A value for m/n can be estimated at tributary junctions. If the tributary (t) joins the principle stream (p) smoothly – without a waterfall – then it can be assumed that incision rate and bedrock resistance are similar for different slopes and drainage areas.

τU = β K-1/n (U1/n-1)(ƒ U1/n – 1)(ƒ U – 1)-1
Where ƒU is the ratio of final to initial rock uplift rate and β is a grouping of constants defined by the equation below.

β = ka-m/n (1 – h m/n)-1 (L1 – h m/n – xc1 – h m/n)   –>  h m/n ≠ 1
and
β = ka-m/n ln(L / xc)-1 (L1-h m/n – xc)   –>  h m/n = 1

t* = τU / tonset
Normalized response time for each watershed. Compare watershed response times (erosional response of drainage network to uplift) with this equation; values of t* are not absolute, rather they are useful as a index to compare watersheds.

……………………..

Factors & Definitions

Ad = drainage area in some equations

Ap = drainage area of principle stream at confluence with tributary

At = drainage area of tributary at confluence with principle stream

Aggrade = Increase in elevation of the channel bed due to deposition of sediment (sediment supply > transport capacity).

A = drainage area at a point along a channel (or hillslope), a proxy for discharge (Q is proportional to A in most mountain watersheds); this is the pixel value from the flow accumulation raster (# of pixels upstream of and flowing into the pixel of interest NOT the entire area of the watershed)

α = fraction of Unit Stream Power apportioned towards breaking bonds in bedrock

B = density of bonds in bedrock; the fewer bonds per unit area, the lower the B value

β = empirical constant or group of constants in the discharge equation (Q = αAβ), determined by statistical regression.

β = channel concavity or the “decay constant of the long profile exponential curve” (Jain et al., 2006); an exponent in the discharge equation (Q = αAβ). Values typical range from 0.6 to 1.0 (Leopold, 1964; Knighton, 1987; Eaton et al., 2002; Flores et al., 2006). Values may be closer to 1.0 in humid climates (Bloschl & Sivapalan, 1997; Eaton et al., 2002).

dz/dt = rate of change in elevation of a channel profile over time (incision of a channel); erosion is assumed to balance rock uplift (dz/dt = 0); note that usage of dz/dt sometimes applies to fall of water down a channel (rate of loss of water elevation) and sometimes the vertical lowering of the channel bed

E = energy required to break bonds between bedrock particles; less bond strength = lower value of E; note different usage of E in different equations

E = erosion rate law governing fluvial incision into rock-bedded channels; E is also called the “stream power law” or “incision rate law”; note different usage of E in different equations

H = depth of channel

h = empirical coefficient

Incision = Decrease in elevation of the channel bed by downcutting (transport capacity > sediment supply).

K = a dimensional erosion coefficient that aggregates several factors including climate and rock type, channel width, flood frequency, channel hydraulics, among others; assumed to be spatially uniform (Braun & Willett, 2013; Lague & Hovius, 2005; Snyder et al., 2000; Whipple & Tucker, 1999; Tinkler, 1998; Howard & Kerby, 1983)

k = efficiency factor of bedrock erosion; k = M/T3; k = kg/s3

ks = channel profile steepness (use same reach endpoints as concavity) is the y-intercept of the same regression line described for θ; ks substitutes for (U/K)1/n; steepness may also be written (U/k)1/n.

ksn = channel steepness index normalized to the reference concavity index θref (Snyder et al., 2000; Kirby & Whipple, 2012; Whittaker, 2012); takes the form of a sublinear power law equation ksn = aPb (D’Arcy & Whittaker, 2013); Units depend on choice of reference concavity (m = 0.9 for θref = 0.45); Normalization facilitates comparison between different rivers; studies have shown a positive correlation of rock uplift (U) and ksn (Snyder et al., 2000; Kirby & Whipple, 2001; Ouimet et al., 2009; Kirby & Whipple, 2012; DiBiase et al., 2010; Lague & Davy, 2003; Cyr et al., 2010; Densmore et al., 2007)

ka = empirical coefficient

ke = a dimensional coefficient

L = total watershed length (m)

m = drainage area exponent, assumed to be spatially uniform; the value of m in steady state stream power models is 0.5 or something close to it.

m/n = -θ if steady state conditions assumed

n = slope exponent; commonly assumed to be 1 for simplicity, assumed to be spatially uniform

θ = channel profile concavity (use same reach endpoints as you did when calculating steepness); θ substitutes for m/n. Values tend to range between 0.4 and 1.0, with commonly used default of 0.6 (see Flint, 1974; Whipple, 2004; Gonga-Saholiariliva et al., 2011).

θref = reference concavity; the mean concavity for all study area watersheds draining to the same range front fault

δ = thickness of a bedrock layer

g = gravitational acceleration, 9.81 m/sec-2

ρ = density of water (1000 kg/m3).

Ω = Stream Power, rate of energy loss to channel bed per unit length (energy per unit time)

Q = discharge (or bankfull discharge), m3/sec.

R = submerged specific gravity of the sediment grains in the critical stream power equation.

R = hydraulic radius of channel, equal to cross sectional area of flow divided by length of the wetted perimeter.

S = local slope of channel (rise/run in meter per meter), usually of the water surface (Knighton, 1999). If using a DEM, elevation values are the discrete slope values (pixel values) of a slope raster along the channel. You must identify “channel” pixels, then the slope of each. If topographic maps (contours) are used, the resolution for determining slope is limited to the contour interval of the map. Field survey of the water surface or the dry channel beds can be made with hand equipment, theodolite total station, or GPS instruments. Aerial and ground-based LiDAR can provides very precise slope data for channels and their beds.

St = slope of tributary at confluence with principle stream

Sp = slope of principle stream at confluence with tributary

Se = Energy slope of the water surface (m/m) approximated by the slope of the water surface or the channel bed.

T = time to break bonds between particles of bedrock, T = BE/αω

τb = mean shear stress imparted by the flow to the channel bed

τc = critical shear stress (bed sediment mobility threshold)

τU = calculated response time for a watershed (erosion response to uplift); values for τU are not absolute, rather they are useful as a index to compare watersheds

t* = normalized response time for a watershed (erosion response to uplift)

t = time

tonset = time of uplift initiation with respect to the watershed position along the fault (fault tip vs. fault center)

U = depth averaged velocity in m/s (mean speed of flow in channel), note different usage of U in different equations

U = rock uplift rate (m/yr), assumed to be spatially uniform (note different usage of U in different equations)

γ = weight of water, 9.81 or 9810 N/m2 or 9807 N/m2 (Reinfelds et al., 2004) or 9800/m3 (Jain et al., 2006) or 9100/m3 (Fonstad, 2003) or 9792 kg/mm3sec2 (Phillips & Desloges, 2013).

V = velocity of flow, the mean depth-averaged flow velocity in m/sec

W = width of channel; not readily found from DEMs; use LiDAR, field measurements, or high-resolution aerial photographs

ω = Unit Stream Power, rate of energy loss to channel bed per area (energy per unit area); can also be written as ω = E/L2T (in SI units) when referring to energy breaking bonds between particles of bedrock.

ωc = critical specific stream power (W/m or N/ms

xc = position of channel head on Slope-Area plot; the flow accumulation, expressed as an area, before a channel begins

xh = empirical coefficient

z = elevation of channel bed

Refs: Specific/Unit/Mean Stream Power
Eaton & Church (2011) Earth Surface Processes and Landforms 36
Ferguson (2005) Geomorphology 70
Flores et al. (2006) Water Resources Research 42
Finlayson et al. (2002) Water Resources Research 10
Golden & Springer (2006, p. 65) Geomorphology 78
Hall et al. (2012, p. 91, Fig. 5) Geomorphology 137
Kleinhans & Van den Berg (2011) Earth Surface Processes and Landforms 36
Knighton (1999) Geomorphology 29
Parker et al. (2011) Geomorphology 126
Phillips & Desloges (2013) Geomorphology
Nanson & Croke (1992) Geomorphology 4
Reinfelds et al (2004, p. 405) Geomorphology 60
Rinaldo et al. (1992) Water Resources Research 28
Van den Berg (1995) Geomorphology 12

Refs: Total/Cross Sectional Stream Power
Fonstad (2003, p. 77) Geomorphology 55
Phillips (1989) Water Resources Bulletin 25
Reinfelds et al (2004, p. 405) Geomorphology 60

Refs: General Stream Power
Anderson & Anderson (2010) Geomorphology: The mechanics and chemistry of landscapes, Cambridge Univ Press, 637 pgs.
Bagnold (1966) USGS Professional Paper 422-I
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Wikipedia: Stream Power Law HERE
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