In estimating erosion, Playfair's law is an empirical relationship that relates the size of a stream to the valley it runs through. It is better defined as a theory rather than law.
In estimating erosion, Playfair's law is an empirical relationship that relates the size of a stream to the valley it runs through. It is better defined as a theory rather than law.
The law states each stream cuts into its own valley, that each valley is proportional in size to its stream and that the stream junctions in the valley are proportional in depth in accordance with the stream level. Although there are models, in the most basic sense it simply says that large, fast streams will be in large valleys and vice versa. This is because steady, heavy flow of water will tug surrounding soil and subsequently erode and deepen the valley it cuts through.
Playfair's law also states that at tributary junctions, tributaries will have the same slope as the main channels. This is referred to by saying that the junctions occur "at grade".
By modeling Playfair's law in the following mathematical scheme, we can find the incision rate of the stream into the valley by following:
In evaluating the case specific parameters, we regard as a parameter that is dependent on the type of land or soil that the stream is penetrating. Also, can be estimated by first assuming that at the junction the law holds over the distance used to determine S and taking their ratios (m/n) at different points, then substitution.
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Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries of physical space.
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The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov and independently in 1968 by James R. Rice, who showed that an energetic contour path integral was independent of the path around a crack.
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In hydrology stream competency, also known as stream competence, is a measure of the maximum size of particles a stream can transport. The particles are made up of grain sizes ranging from large to small and include boulders, rocks, pebbles, sand, silt, and clay. These particles make up the bed load of the stream. Stream competence was originally simplified by the “sixth-power-law,” which states the mass of a particle that can be moved is proportional to the velocity of the river raised to the sixth power. This refers to the stream bed velocity which is difficult to measure or estimate due to the many factors that cause slight variances in stream velocities.
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 and basin hydrology 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:
Plasticity theory for rocks is concerned with the response of rocks to loads beyond the elastic limit. Historically, conventional wisdom has it that rock is brittle and fails by fracture while plasticity is identified with ductile materials. In field scale rock masses, structural discontinuities exist in the rock indicating that failure has taken place. Since the rock has not fallen apart, contrary to expectation of brittle behavior, clearly elasticity theory is not the last word.
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.
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:
| Large-scale features | |
|---|---|
| Alluvial rivers | |
| Bedrock river | |
| Bedforms | |
| Regional processes | |
| Mechanics | |
