The term stream power law describes a semi-

empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...

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

river A river is a natural flowing watercourse, usually freshwater Fresh water or freshwater is any naturally occurring liquid or frozen water containing low concentrations of dissolved salts and other total dissolved solids. Although the ...

into its bed. These combine equations describing conservation of water mass and momentum in streams with relations for channel

hydraulic geometry Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counte ...

(width-discharge scaling) and

basin Basin may refer to: Geography and geology * Depression (geology) ** Back-arc basin, a submarine feature associated with island arcs and subduction zones ** Debris basin, designed to prevent damage from debris flow ** Drainage basin (hydrology), ...

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: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...

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 proportional relative change in the other quantity, independent of the initial size of those quantities: one q ...

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 water flow or wind) that removes soil, rock, or dissolved material from one location on the Earth's crust, and then transports it to another location where it is deposited. Erosion is di ...

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 be ...

erodibility Erodability (or erodibility) 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 mechani ...

climate Climate is the long-term weather pattern in an area, 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 meteorologica ...

sediment Sediment is a naturally occurring material that is broken down by processes of weathering and erosion, and is subsequently transported by the action of wind, water, or ice or by the force of gravity acting on the particles. For example, sa ...

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, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediat ...

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 field 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 al ...

, 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 slope, such as a waterfall or lake. Knickpoints reflect different conditions and processes on the river, often caused by previous ...

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 interval (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 be ...

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

References

Geomorphology {{geomorph-stub