Stream power law

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The term stream power law describes a semi-empirical family of equations used to predict the rate of erosion of a river into its bed. These combine equations describing conservation of water mass and momentum in streams with relations for channel hydraulic geometry (width-discharge scaling) and basin hydrology (discharge-area scaling) and an assumed dependency of erosion rate on either unit stream power or shear stress on the bed to produce a simplified description of erosion rate as a function of power laws of upstream drainage area, A, and channel slope, S:

where E is erosion rate and K, m and n are positive. [1] The value of these parameters depends on the assumptions made, but all forms of the law can be expressed in this basic form.

The parameters K, m and n are not necessarily constant, but rather may vary as functions of the assumed scaling laws, erosion process, bedrock erodibility, climate, sediment flux, and/or erosion threshold. However, observations of the hydraulic scaling of real rivers believed to be in erosional steady state indicate that the ratio m/n should be around 0.5, which provides a basic test of the applicability of each formulation. [2]

Although consisting of the product of two power laws, the term stream power law refers to the derivation of the early forms of the equation from assumptions of erosion dependency on stream power, rather than to the presence of power laws in the equation. This relation is not a true scientific law, but rather a heuristic description of erosion processes based on previously observed scaling relations which may or may not be applicable in any given natural setting.

The stream power law is an example of a one dimensional advection equation, more specifically a hyperbolic partial differential equation. Typically, the equation is used to simulate propagating incision pulses creating discontinuities or knickpoints in the river profile. Commonly used first order finite difference methods to solve the stream power law may result in significant numerical diffusion which can be prevented by the use of analytical solutions [3] or higher order numerical schemes . [4]

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<span class="mw-page-title-main">Geomorphology</span> Scientific study of landforms

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<span class="mw-page-title-main">Playfair's law</span>

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<span class="mw-page-title-main">Stream power</span>

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.

<span class="mw-page-title-main">Bedrock river</span> Type of river

A bedrock river is a river that has little to no alluvium mantling the bedrock over which it flows. However, most bedrock rivers are not pure forms; they are a combination of a bedrock channel and an alluvial channel. The way one can distinguish between bedrock rivers and alluvial rivers is through the extent of sediment cover.

In geology, degradation refers to the lowering of a fluvial surface, such as a stream bed or floodplain, through erosional processes. Degradation is the opposite of aggradation. Degradation is characteristic of channel networks in which either bedrock erosion is taking place, or in systems that are sediment-starved and are therefore entraining more material than they are depositing. When a stream degrades, it leaves behind a fluvial terrace. This can be further classified as a strath terrace—a bedrock terrace that may have a thin mantle of alluvium—if the river is incising through bedrock. These terraces may be dated with methods such as cosmogenic radionuclide dating, OSL dating, and paleomagnetic dating to find when a river was at a particular level and how quickly it is downcutting.

<span class="mw-page-title-main">Valley network (Mars)</span> Branching networks of valleys on Mars

Valley networks are branching networks of valleys on Mars that superficially resemble terrestrial river drainage basins. They are found mainly incised into the terrain of the martian southern highlands, and are typically - though not always - of Noachian age. The individual valleys are typically less than 5 kilometers wide, though they may extend for up to hundreds or even thousands of kilometers across the martian surface.

<span class="mw-page-title-main">Pothole (landform)</span> Natural bowl-shaped hollow carved into a streambed

In Earth science, a pothole is a smooth, bowl-shaped or cylindrical hollow, generally deeper than wide, found carved into the rocky bed of a watercourse. Other names used for riverine potholes are pot, (stream) kettle, giant's kettle, evorsion, hollow, rock mill, churn hole, eddy mill, and kolk. Although somewhat related to a pothole in origin, a plunge pool is the deep depression in a stream bed at the base of a waterfall. It is created by the erosional forces of turbulence generated by water falling on rocks at a waterfall's base where the water impacts. Potholes are also sometimes referred to as swirlholes. This word was created to avoid confusion with an English term for a vertical or steeply inclined karstic shaft in limestone. However, given widespread usage of this term for a type of fluvial sculpted bedrock landform, pothole is preferred in usage to swirlhole.

Erodability is the inherent yielding or nonresistance of soils and rocks to erosion. A high erodability implies that the same amount of work exerted by the erosion processes leads to a larger removal of material. Because the mechanics behind erosion depend upon the competence and coherence of the material, erodability is treated in different ways depending on the type of surface that eroded.

<span class="mw-page-title-main">River incision</span>

River incision is the narrow erosion caused by a river or stream that is far from its base level. River incision is common after tectonic uplift of the landscape. Incision by multiple rivers result in a dissected landscape, for example a dissected plateau. River incision is the natural process by which a river cuts downward into its bed, deepening the active channel. Though it is a natural process, it can be accelerated rapidly by human factors including land use changes such as timber harvest, mining, agriculture, and road and dam construction. The rate of incision is a function of basal shear-stress. Shear stress is increased by factors such as sediment in the water, which increase its density. Shear stress is proportional to water mass, gravity, and WSS:

Hillslope evolution is the changes in the erosion rates, erosion styles and form of slopes of hills and mountains over time.

References

  1. Whipple, K.X. and Tucker, G.E., 1999, Dynamics of the stream-power incision model: Implications for height limits of mountain ranges, landscape response timescales, and research needs, J. Geophys. Res., v.104(B8), p.17661-17674.
  2. Whipple, K.X., 2004, Bedrock Rivers and the Geomorphology of Active Orogens, Annu. Rev. Earth Planet. Sci., v.32, p.151-85.
  3. Royden, Leigh; Perron, Taylor (2013-05-02). "Solutions of the stream power equation and application to the evolution of river longitudinal profiles". J. Geophys. Res. Earth Surf. 118 (2): 497–518. Bibcode:2013JGRF..118..497R. doi:10.1002/jgrf.20031. hdl: 1721.1/85608 . S2CID   15647009.
  4. Campforts, Benjamin; Govers, Gerard (2015-07-08). "Keeping the edge: A numerical method that avoids knickpoint smearing when solving the stream power law". J. Geophys. Res. Earth Surf. 120 (7): 1189–1205. Bibcode:2015JGRF..120.1189C. doi: 10.1002/2014JF003376 . S2CID   128587259.