The term stream power law describes a semi-

empirical Empirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. There is no general agreement on how t ...

family of equations used to predict the rate of erosion of a

river A river is a natural stream of fresh water that flows on land or inside Subterranean river, caves towards another body of water at a lower elevation, such as an ocean, lake, or another river. A river may run dry before reaching the end of ...

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 Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...

on the bed to produce a simplified description of erosion rate as a function of

power laws In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent: one quantity va ...

of upstream drainage area, ''A'', and channel slope, ''S'': :

E = K A^m S^n

where ''E'' is erosion rate and ''K'', ''m'' and ''n'' are positive. 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 Erosion is the action of surface processes (such as Surface runoff, water flow or wind) that removes soil, Rock (geology), rock, or dissolved material from one location on the Earth's crust#Crust, Earth's crust and then sediment transport, tran ...

process,

bedrock In geology, bedrock is solid rock that lies under loose material ( regolith) within the crust of Earth or another terrestrial planet. Definition Bedrock is the solid rock that underlies looser surface material. An exposed portion of bed ...

erodibility Erodability (or erodibility) is the inherent yielding or nonresistance of soils and rocks to erosion. A high erodibility implies that the same amount of work exerted by the erosion processes leads to a larger removal of material. Because the mechan ...

climate Climate is the long-term weather pattern in a region, typically averaged over 30 years. More rigorously, it is the mean and variability of meteorological variables over a time spanning from months to millions of years. Some of the meteoro ...

sediment Sediment is a solid material that is transported to a new location where it is deposited. It occurs naturally and, through the processes of weathering and erosion, is broken down and subsequently sediment transport, transported by the action of ...

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. 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 A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...

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 In the fields of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a ...

, more specifically a

hyperbolic partial differential equation In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n - 1 derivatives. More precisely, the Cauchy problem can ...

. Typically, the equation is used to simulate propagating incision pulses creating discontinuities or

knickpoint In geomorphology, a knickpoint or nickpoint is part of a river or channel where there is a sharp change in channel bed slope, such as a waterfall or lake. Knickpoints reflect different conditions and processes on the river, often caused by pre ...

s in the river profile. Commonly used first order

finite difference methods In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time domain (if applicable) are dis ...

to solve the stream power law may result in significant

numerical diffusion Numerical diffusion is a difficulty with computer simulations of continua (such as fluids) wherein the simulated medium exhibits a higher diffusivity than the true medium. This phenomenon can be particularly egregious when the system should not b ...

which can be prevented by the use of analytical solutions or higher order numerical schemes .

References

Geomorphology {{geomorph-stub