Dimensional analysis permits quantitative geomorphic studies to be placed on a sound geometrical and mechanical basis Form elements of fluvially dissected landmasses are first analyzed according to dimension. Stream length, relief, length of overland flow, and basin perimeter have the dimension of length L. Drainage density, texture ratio, and curvature of profile have an inverse length dimension L−1. Areal measures and volumes have the dimensions of length squared L2 and length cubed L3, respectively. Dimensionless parameters include stream-order number, stream azimuth, ground-slope angle, and channel gradient. Combinations of dimensional elements produce dimensionless numbers, such as stream-length ratio, basin circularity ratio, ruggedness “number, and hypsometric integral, which provide descriptive indices of the terrain, irrespective of scale.
Corrsin's dimension space is a useful graphic device for clarifying the dimensions of physical properties as unique vectors extending from an origin to appropriate points in space with respect to three orthogonal coordinates scaled in integer powers.
Application of dimensional analysis and the Pi Theorem to drainage density as a function of runoff intensity, an erosion proportionality factor, relief, fluid density and viscosity, and gravity yields four dimensionless groups: a ruggedness number, an erosion-intensity number (here named the Horton number), a Reynolds number, and a Froude number.
Complete geometrical similarity of landforms in two regions exists when all corresponding linear dimensions are in the same scale ratio, and all corresponding dimensionless numbers are the same. As an illustration, approximate similarity in planimetric form, but not in relief and slope, is established between small areas of mountainous terrain in the Verdugo Hills, California, and the Great Smoky Mountains, North Carolina. Examples from the Clinch Mountain area of western Virginia illustrate a case of extremely close similarity in landforms developed on different rock types and a case of striking dissimilarity where structural and lithologic controls are strong.
Accelerated land erosion, in which gullying results in badlands, is interpreted as a drainage-density transformation in response to sharp increases in the Horton number. Despite a many-fold increase in drainage density, the geometry number tends to be conserved by reduction of relief and increase of slope. Geometrical similarity is preserved in planimetric aspect but not in vertical aspect. The resultant system reaches a new steady state capable of exporting greatly increased load because of steepened valley-side and channel gradients.