Basin Perimeter Research Articles

The energetics of large-scale internal wave degeneration in lakes

Field observations in lakes, where the effects of the Earth's rotation can be neglected, suggest that the basin-scale internal wave field may be decomposed into a standing seiche, a progressive nonlinear surge and a dispersive solitary wave packet. In this study we use laboratory experiments to quantify the temporal energy distribution and flux between these three component internal wave modes. The system is subjected to a single forcing event creating available potential energy at time zero (APE). During the first horizontal mode one basin-scale wave period (T i ), as much as 10 % and 20 % of the APE may be found in the solitary waves and surge, respectively. The remainder is contained in the horizontal mode one seiche or lost to viscous dissipation. These findings suggest that linear analytical solutions, which consider only basin-scale wave motions, may significantly underestimate the total energy contained in the internal wave field. Furthermore, linear theories prohibit the development of the progressive nonlinear surge, which serves as a vital link between basin-scale and sub-basin-scale motions. The surge receives up to 20 % of the APE during a nonlinear steepening phase and, in turn, conveys this energy to the smaller-scale solitary waves as dispersion becomes significant. This temporal energy flux may be quantified in terms of the ratio of the linear and nonlinear terms in the nonlinear non-dispersive wave equation. Through estimation of the viscous energy loss, it was established that all inter-modal energy flux occurred before 2T i ; the modes being independently damped thereafter. The solitary wave energy remained available to propagate to the basin perimeter, where although it is beyond the scope of this study, wave breaking is expected. These results suggest that a periodically forced system with sloping topography, such as a typical lake, may sustain a quasi-steady flux of 20 % of APE to the benthic boundary layer at the depth of the metalimnion.

The Atmospheric Hydrologic Cycle over the Arctic Basin from Reanalyses. Part I: Comparison with Observations and Previous Studies*

The atmospheric moisture budget is evaluated for the region 70°N to the North Pole using reanalysis datasets of the European Centre for Medium-Range Weather Forecasts (ECMWF; ERA: ECMWF Re-Analysis) and the collaborative effort of the National Centers for Environmental Prediction (NCEP) and the National Center for Atmospheric Research (NCAR). For the forecast fields of the reanalyses, the ERA annually averaged P − E (precipitation minus evaporation/sublimation) field reproduces the major features of the basin perimeter as they are known, while the NCEP–NCAR reanalysis forecast fields contain a spurious wave pattern in both P and E. Comparisons between gauge data from Soviet drift camp stations and forecast P values of the reanalyses show reasonable agreement given the difficulties (i.e., gauge accuracy, translating location). When averaged for 70°–90°N, the ERA and NCEP–NCAR forecast P − E are similar in the annual cycle. Average reanalysis forecast values of E for the north polar cap are found to be 40% or more too large based on comparisons using surface latent heat flux climatologies. Differences between a synthesized average moisture flux across 70°N from rawinsonde data of the Historical Arctic Rawinsonde Archive (HARA) and the reanalysis data occur in the presence of rawinsonde network problems. It is concluded that critical deficiencies exist in the rawinsonde depiction of the summertime meridional moisture transport. However, it remains to be seen whether the rawinsonde estimate can be rectified with a different method. For 70°–90°N, annual moisture convergence (P − E) values from the ERA and NCEP–NCAR are very similar; for both reanalyses, annual P − E values obtained from forecast fields are much lower than those obtained from moisture flux convergence by about 60%, indicating severe nonclosure of the atmospheric moisture budget. The nonclosure primarily results from anomalously large forecast E values. In comparison with other studies, reanalyses moisture convergence values are much more reasonable. A synthesis of the reanalysis moisture convergence values and more recent studies yields a value of 18.9 ± 2.3 cm yr−1 for the north polar cap.

Neogene tectonic evolution and exhumation of the southern Ecuadorian Andes: a combined stratigraphy and fission-track approach

Coastal marine and continental sedimentary facies of Middle to Late Miocene age are exposed in the Andes of southern Ecuador (Cuenca, Girón–Santa Isabel, Loja, Malacatos–Vilcabamba and Catamayo–Gonzanamá Basins). The chronostratigraphy of the basin series was established by zircon fission-track dating on a total of 120 tephra layers. Subsequently, the timing of tectonic events was estimated through the well-dated stratigraphic sequences and intervening unconformities. Sedimentation from ≈15 to 9 Ma (termed Pacific Coastal Stage) was dominantly of coastal marine type, extending over an area far greater than the present basin perimeters. It ended when a period of east–west-oriented compression at ≈9.5–8 Ma exhumed the region, and sedimentation was then restricted to smaller basins (termed Intermontane Stage). These Late Miocene continental sediments were for the first time sourced from the west in the rising Western Cordillera. Apatite fission-track analysis was applied to some of the tephras in the Cuenca Basin and also to the older (Eocene, 42–35 Ma) Quingeo Basin series in order to quantify the basin histories with respect to timing and amount of burial and later exhumation. In the Quingeo Basin burial of the oldest sediments reached temperatures of ∼100°C at 18 Ma, when they started to cool down during a period of exhumation. This process preceded the Pacific Coastal Stage development of the other basins. In the Cuenca Basin, the oldest sediments were buried to temperatures of ca. 120°C by 9 Ma, when a period of inversion began and a phase of erosion was dominant. This timing correlates well with that estimated from structural evidence. At ca. 6 Ma the cooling rate slowed down and maybe even reverted to a small increase in temperature until 3 Ma, when the final stages of exhumation took place. Assuming a geothermal gradient of 35°C/km, total uplift for this part for Ecuador is about 6100 m over the last 9 million years. Assuming a steady state continuous movement, this means a mean rock uplift rate of ∼0.7 mm/yr and a surface uplift of 0.3 mm/yr to the Present.

Review: “The Hydrogeology of the Chalk of North-West Europe”

The chalk deposits of Late Cretaceous and Early Paleocene age in and around the present North Sea Basin form an important regional aquifer in the shallow subsurface on the basin perimeter, and a major hydrocarbon reservoir in the deeper parts of the basin. Currently, about 8.5 million cubic meters of water are pumped daily from the chalk. About 80 percent of the withdrawal is in southern England and northern France. This book provides an overview of the regional hydrogeology of the chalk, with primary emphasis on the fresh-water reservoir. The first six chapters of the book discuss topics applicable to the aquifer as a whole. Chapter 1, "The Making of an Aquifer," briefly describes the regional structure and stratigraphy and the diagenesis and modification of the chalk into a regionally extensive aquifer. Chapter 2, "The Formation and Diagenesis of the Chalk," poses the question, "Why so much chalk?" It explains how the combination of a rise in sea level, lack of erosion in nearby land areas, and higher temperatures and higher C02 in the atmosphere during Late Cretaceous time led to the extensive accumulation of chalk ooze. The resulting rock consists almost entirely of skeletal fragments of algae composed of lowmagnesium calcite. Chapter 3, "The Chalk as an Aquifer," describes the dual-porosity nature of the chalk, which generally consists of a highly porous matrix of low permeability and a fracture network that contains both tight tectonic fractures and highly permeable karstic fissures. Flow in the unsaturated and saturated zones is described in detail, and downgradient changes in the groundwater chemistry are discussed. The other three introductory chapters discuss simulation of flow and transport, engineering geology, and groundwater pollution in the chalk. The middle section of the book provides descriptions of the chalk aquifer in France, Belgium, the Netherlands, Denmark, Sweden, and the United Kingdom. These six chapters generally describe the aquifer extent, stratigraphy and structure, aquifer properties, groundwater levels and potentiometric surfaces, recharge and discharge characteristics, natural water chemistry and pollution, and current groundwater development in each country. The section on France provides a detailed discussion (15 pages) that relates groundwater levels and flow in the chalk of the Paris Basin to geologic structures, topography, and karstic networks. Ground pollution by nitrates and pesticides also receives a lot of attention. The chapter on Belgium consists largely of descriptions of the three major chalk aquifers: the Hesbaye aquifer, the Mons Basin, and the confined aquifers of northern Belgium, with emphasis on modeling approaches. The chapter describes a complex, 5-layer finite-element model of the Hesbaye aquifer, incorporating pumping from wells and galleries, faults, and various geologic unit boundaries. The stratigraphy and lithology of the chalk in the Netherlands are described for the two separate areas of occurrence in the northern and southern parts of the country. Detailed hydrogeologic information is presented only for the southern part of the Netherlands, where the aquifer is near surface and characterized by a matrix of high porosity and very low permeability; fractures and dissolution openings provide high secondary permeability. The chapter on Denmark contains brief summaries of the structure, stratigraphy, karst features, and aquifer properties; the principal focus of the chapter is on hydrochemistry, including natural chemical constituents and pollution problems. In Sweden, the occurrence ofthe chalk is limited to four areas in the southernmost part of the country. The structural setting, stratigraphy, aquifer properties, karstic features, water chemistry, and groundwater development are described for these areas. The chalk

Kinematics of arc‐continent collision in the eastern Central Range of Taiwan inferred from geomorphic analysis

Active arc‐continent collision in Taiwan is an example of a typical collisional orogen, with a deformed continental margin (Foothills Belt, Hsueshan Range, and Central Range) separated from an accreted island arc (Coastal Range) by a collapsed forearc basin (Longitudinal Valley). The geomorphology of the eastern Central Range reflects spatial and temporal variations in uplift rate and pattern along the orogen during late Quaternary time. Digitized drainage basin perimeters, stream channels, and the mountain‐piedmont junction of a 200‐km‐long segment of the eastern Central Range provide data for calculating quantitative geomorphic parameters. Statistical analyses of these parameters define four domains of distinct landscape character. The northern domain extends northward along the coast from the northern end of the Longitudinal Valley (LV) and has landscape characteristics suggesting uplift near the mountain front. The central two domains are adjacent to the LV and show evidence of uplift concentrated in the interior of the range, away from the front. The boundary between the two central domains occurs near the middle of the LV. The southern domain extends southward along the coast from the southern end of the LV and has characteristics suggesting a transient response to increased uplift rate. Analysis of fluvial terrace profiles in the Central Range adjacent to the southern half of the LV indicates a change in uplift pattern through time, with migration of maximum uplift rate from the mountain front to the interior of the range. This change probably occurred just prior to late Pleistocene time.

THE ROLE OF PERIMETER SHAPE IN ESTIMATING ANNUAL RUNOFF FROM SMALL SIERRA NEVADA BASINS

Perimeter shape of drainage basins is evaluated as a control of annual runoff depth from small Sierra Nevada basins underlain by granitic rocks. In contrast to three perimeter shape ratios, none of which contributes a statistical explanation of annual runoff depth, three terms of a Fourier expansion of the basin perimeter substantially improve the estimating model. When added to annual precipitation, the first, third, and seventh Fourier amplitudes increase statistical explanation by 15 percent and prediction accuracy by 10 percent. These improvements are much greater than corresponding gains atributable to other morphological variables. [Key words: basin runoff, basin shape, Sierra Nevada.]

Drainage basin perimeters: a fractal significance

Drainage basin boundaries are irregular curves with many scales of crenulation, defying precise measurement of basin perimeter. Richardson “divider” analysis of six drainage basin outlines from southern Indiana, and of an additional six from larger U.S. basins, suggests that basin boundaries are fractal or near-fractal shapes. Although the basins differ significantly in physiography, the fractal dimension values obtained fit within the narrow range 1.06–1.12, indicating that while basins may differ greatly in overall shape, they are very similar in terms of smaller-scale boundary irregularity. Techniques of measuring basin perimeter should be modified in recognition of these facts. The consistent fractal or near-fractal geometric characteristics of basin boundaries give a basis for extrapolation of perimeter values to appropriate levels of resolution. For most basin morphometry applications, however, the objective should be to make comparison among basins using measurements taken at the same degree of resolution relative to basin size, rather than identical resolution.

Variability in two deep coastal basins (Santa Monica and San Pedro) off southern California

Time series of midbasin hydrographic samples and of an array of moored temperature‐velocity sensors are used to examine subtidal to interannual scale variability in two ∼900‐m‐deep basins off the southern California coast. The deepest sill that allows entry into the basins is at 740 m, and the two basins, Santa Monica and San Pedro, are separated by a sill about 40 m high. The mean flow 40 m above and 60 m below the 740‐m sill depth is in the counterclockwise direction and the magnitude is ∼0.5–1.0 cm s−1. The data indicate that the waters at deeper levels of the basins are neither quiescent, at one extreme, nor are they filled with random noise, at the other extreme. Rather, at all scales examined, the variability is significant and much of it is spatially coherent. The amplitude of the subtidal velocity fluctuations is ∼4 cm s−1, almost an order of magnitude greater than that of the mean flow. The amplitude of the subtidal temperature fluctuations is ∼0.1°C at 700 m and ∼0.05°C at 800 m. The temperature variance is roughly a factor of 2 smaller at midbasin than on the basin perimeter. This is not the case with the velocity variance, which is generally greatest at midbasin and on the coastal edges of both basins. Much of the subtidal variability in both temperature and velocity appears to be associated with topographic waves that travel counterclockwise around the pair of basins. The observed period of the waves (∼10–20 days) is roughly consistent with that predicted by a barotropic model of free waves in an elliptical basin (Mysak, 1985). Comparison of time series of currents in the upper (<200 m) water column with those in the lower water column suggests that the waves are forced by the circulation in the upper water column, which is directed over (rather than around) the basins. Seasonal variability was observed in both temperature and velocity fields as deep as 800 m (i.e., below the depth of the deepest sill) along the perimeters of both basins and at midbasin in Santa Monica but not in San Pedro. Temperature increases and salinity and density decrease from spring to fall, consistent with the seasonal development of the poleward flowing California Countercurrent in the upper water column. The seasonal cycle of water properties has significant spatial variability over the basins: the amplitude is a factor of 2 greater on the seaward side of Santa Monica basin than at midbasin and on its coastal edges, and on the edges of San Pedro basin. The variance of the subtidal velocity and temperature fields also exhibit a seasonal signal: maximum variance occurs during fall; minimum variance occurs during spring. Interannual variability dominates the 4‐year time series of water properties of the near‐bottom water in each basin, and differs somewhat between the two basins. The interannual variability is dominated by events associated with renewal of basin bottom water. Two such events were identified: a minor one in 1987; and a major one in 1988, each beginning in May of that year. In both cases, the actual renewal occurs in less than a month, and the most rapid return to near‐normal conditions occurs over a period of about 3–5 months.

Alluvial sedimentation patterns in the Munster Basin, Ireland

ABSTRACTPost‐Caledonian southern Ireland witnessed the development of a NE‐SW orientated half‐graben known as the Munster Basin. More than 7 km of non‐marine sediments accumulated in the basin during the late Middle and Late Devonian. Marine conditions became established in the southern part of the basin at the end of the Devonian. In this paper, a model for the evolving style of sedimentation in the basin and its periphery is constructed with the aim of identifying the major factors which controlled sedimentation patterns and the architecture of the basin fill.The depositional history of the basin is considered in terms of four successive episodes. During Episode I, gravelly alluvial fans flanked upland areas around the northeastern and northern basin perimeter. These graded southwestwards to a floodplain dominated by sheet‐floods. In the western part of the basin, the first of three major fluvial influxes into the basin commenced. During Episode II, the first influx developed into a large sandy braided complex. The sediment was derived from a distant source area located to the north and west of the basin and was transported diagonally across the basin towards the southeast. Episode III witnessed a second influx which drained into the basin from the northeast and north. River channels were of low sinuosity and graded distally to an ephemeral playa lake. Episode IV was marked by a third fluvial influx from the west and northwest. This was confined to the southern half of the basin and drainage was directed towards the east. The fluvial distributaries were flanked by permanently flooded overbank areas. This influx coincided with the first marine transgression which advanced westwards. The end of Episode IV coincided with the beginning of the Carboniferous and was marked by a major marine transgression.Sediment input to the basin was influenced by stable areas occupied by granitic plutons on either side of the basin and a southward downthrowing fault along its northern margin. The drainage direction was principally controlled by E‐W trending within‐basin faults and an E‐W trending stable area located to the south. The basin was fundamentally of the axial transport type, the main drainage having been directed towards the east though there was also a strong lateral influx from the north, northwest and northeast. Stable areas around the, basin periphery resulted in either no sediment preservation or sequences of multistorey channel deposits while thick sequences dominated by fine‐grained floodplain or overbank deposits characterized areas of higher subsidence rate within the basin. Movement on the northern basin‐margin fault was probably the major cause of the first fluvial influx, while regional subsidence of the basin and its northern periphery resulted in the second influx. The third influx was a response to local subsidence in the southern part of the Munster Basin. This also contributed to the simultaneous westward marine transgression in this area towards the end of the Devonian. Source area denudation and retreat in association with a sea‐level rise were ultimately responsible for terminating the alluvial regime in the Munster Basin.

Automatic delineation of drainage basins within digital elevation data using the topographic primal sketch

A new technique for automatic delineation of drainage basin perimeters within digital elevation data is described which applies the topographic primal sketch to digital terrain data, and is much more natural and much more efficient than usual methods that rely upon moving a window over the array of data. This method is able to find drainage channels, mark water flow direction, and recover drainage basin areas. Results appear to be comparable to drainage patterns which can be interpreted visually from contours of the elevation data.

Optimal Jet‐Mixer for Shallow Basins

A series of experimental tests are carried out to aid in the optimal design of a jet‐mixer for a shallow circular basin. Mixing times for single and double jets with various orientations are investigated to determine the best arrangement. The best single jet is one directed along the diameter of the basin. A tangential jet near the basin perimeter has the longest mixing time. The best overall arrangement consists of tangential jet and 45° jet at a radius of 0.4 of the basin radius. An optimal design procedure is suggested.

Late Quaternary evolution of Lake Carpentaria on the Australia‐New Guinea continental shelf

During the Late Pleistocene, the large and shallow Lake Carpentaria occupied part of the continental shelf linking northern Australia and New Guinea. The former lake lay within the modern Gulf of Carpentaria, between the —53 m isobath and the deepest part of the embayment at — 67 m, and may have been as much as 500 km long, 250 km wide and 15 m deep. The basinal feature is separated from the outer margin of the continental shelf in the northwest by an area of shallower sea floor with a maximum water depth of about — 53 m and referred to as the Arafura Sill. We have investigated the sedimentary record of the shallow substrate, both within and beyond the basin perimeter, by using surficial sediment samples, piston cores, and continuous seismic reflection profiling. Sediments interpreted to be of lacustrine origin occur only within the enclosed basin, and have been covered by a blanketing marine sandy mud deposit approximately 1 m thick. Beneath the lacustrine mud and sandy mud is a cohesive clay and sandy c...

Fault Severed Aquifers Along the Perimeters of Wyoming Artesian Basins

ABSTRACTThe mountain uplifts which border the major artesian basins of Wyoming are asymmetric antiforms bounded on one flank by large displacement Laramide thrust faults. These thrusts sever the hydraulic continuity of aquifers, thereby creating separate circulation systems in the hanging wall and foot wall blocks. Fault severing of the Paleozoic aquifers can be identified by (1) potentiometric discontinuities across the faults, (2) water quality contrasts across the faults, and (3) geothermally heated waters in the foot wall blocks.Isolated but active circulation systems develop in the hanging wall blocks in which good permeabilities and good quality water prevail. In contrast the foot walls are generally characterized by poor permeabilities and poor water qualities.Fault severing of the Paleozoic aquifers in the Wyoming foreland province is a significant factor because nearly half of the basin perimeters are fault severed. The result is that exploration for large‐volume, good‐quality supplies must focus on the hanging wall blocks in the fault severed environments.

Analysis of length distribution of drainage basin perimeter

To establish a theoretical base for the study of the length distribution of basin perimeters, the paper introduces a descriptive model of the topology of interlocking channel and ridge networks. Assuming topological randomness within and between both, the expected number of links of basin perimeters is derived; for large basin magnitudes n, it approximates a square root function in n. Observed link numbers of perimeters deviate significantly, showing a 0.69 regression exponent for their growth rate relative to the basin magnitude rather than the expected value of 0.5. The spatial constraint of possible perimeter/(area;)½ proportions as defined by the circle is translated into a corresponding topological constraint but fails to provide a sufficient explanation. The paper then explores the possibility that the relatively large length of the perimeter reflects the basin elongation which, following Hack, might be linked to the length of the mainstream. Although basin perimeter, elongation, and mainstream length are highly correlated and the elongation axis is oriented to the outlet in two‐thirds of the sample basins, the data indicate that the mainstream link number does not account for the basin elongation, nor does it account for the number of links of the basin perimeter.

An evaluation of landscape units for watershed similarity analyses

The landscape unit approach to land classification was applied through the use of aerial imagery interpretation to Guadalupe Mountains National Park, Texas. The delineated units were evaluated for uniqueness on the basis of their percentage composition in three attributes: vegetative cover, slope class and geologic parent material. Major watersheds within the park were then characterized in terms of their percentage composition with respect to the delineated landscape units. Simultaneously, watersheds were characterized using traditional hydrologic inventory and description data such as: areal extent, Miller's circularity ratio, basin perimeter, drainage density, main channel length, mean channel slope, maximum—minimum elevations, vegetative composition, geologic composition and slope class composition. Using the two different data bases (i.e. traditional approach variables and landscape unit composition) a similarity index computer program was then employed to examine all possible pairs of watersheds and to determine the twenty most similar pairs in each case. These pairs were further examined to develop clusters of highly similar watersheds. Comparison of the similar watershed lists and the clusters developed from the two data bases indicated very favorable results. The time and effort to obtain clusters of highly similar watersheds by means of the hydrologic variable analyses approach exceeded the landscape unit approach by a factor of ten. This reduction in cost coupled with the favorable comparison of the final results indicates a potential for increased use of the landscape unit analysis approach to identify drainage basins which could be expected to respond more or less uniformly to particular management prescriptions.

Subsidence of the Sverdrup Basin, Canadian Arctic Islands

Sverdrup Basin subsidence is separated into three phases: late Paleozoic (330 to 225 m.y. B.P.), early Mesozoic (225 to 124 m.y. B.P.), and late Mesozoic (124 to 74 m.y. B.P.). Each phase begins with high subsidence rates that decrease exponentially with time. During periods of rapid basin subsidence, regions adjacent to the basin perimeter experience mild uplift that migrates inward toward the basin center over a 10- to 30-m.y. period. Most of the observed basin subsidence (at least 70 percent) probably results from lithospheric response to loading of an initial depression. For the late Paleozoic and most of the early Mesozoic phases, observed subsidence and peripheral uplift relations, together with the pattern of exponential decay constants determined from subsidence curves, are consistent with the loading response of a lithosphere modeled as a viscoelastic beam. Late Mesozoic subsidence, because of thermal and structural events within the basin, presents no clear picture of lithospheric loading response. Late Paleozoic subsidence may have been initiated by thermal contraction of the lithosphere, whereas early Mesozoic subsidence probably resulted from downdropping of a graben block beneath the basin and late Mesozoic subsidence from downdropping centered along an axis collinear with the north-trending Boothia Arch exposed to the south of Sverdrup Basin. Pronounced increases in subsidence rates, mafic intrusions, and cessation of north-derived sediment in Sverdrup Basin at the beginning of Mesozoic time suggest that the opening of the Amerasian Basin of the Arctic Ocean by rifting and sea-floor spreading may have begun in Early Triassic time.

On Coastal Jets and Upwelling in Bounded Basins

Abstract The relationship between coastal upwelling and coastal long-short currents is studied in a simple model of a continuously stratified fluid in a closed basin on the f-plane. It is shown that the coastal long-shore current at any point on the basin's perimeter is determined primarily by the components of the long-shore stress with the largest long-shore scales, in particular from the perimeter average of the long-shore stress. In addition, the depth structure of the upwelling itself is shown to be dependent on the long-shore scale. It is concluded that the relationship between upwelling and long-shore currents is non-local, that mass balance is not achieved locally in planes normal to the coast, and that the relationship between onshore flows and long-shore currents is scale-dependent. An intrinsic long-shore length scale is found separating two regimes of flow with distinctly different relations between the wind stress, onshore, and long-shore flows. Since both regimes are present simultaneously f...

Factor analysis of drainage basin properties: Evidence for stream abstraction accompanying the degradation of relief

Factor analyses are carried out on five parameters measured in third order basins in five landscapes in eastern Australia: basin area, stream length, basin relief, basin perimeter, and stream number. A comparison of the factor solutions for the five landscapes suggests that departures from topological randomness are associated with the evolutionary decline of relief. Regressions between the geometric mean bifurcation ratio and relief indicate that very small headwater streams have been abstracted from the drainage networks in the two landscapes where relief has been erosionally reduced below 200 feet.

Geometric Properties of Mature Drainage Systems and Their Representation in an E4Phase Space

Topographic texture is measured by two distinct parameters: drainage density (D) and channel frequency (F). In maturely dissected regions the functional relation between D and F is (1) F = 0.694D2. The dimensionless ratio F/D2 varies inversely with valley-side slope (θ) and basin relief (R), area and channel length held constant, but is not affected by basin circularity or basin perimeter (P). The ratio F/D2 is interpreted as a measure of the completeness with which a channel system fills a basin outline, for given values of N (number of channels). For an ideal basin of 1 square mile area, it is postulated that the equation N = 0.694L2 (L = total channel length) is a growth model, although it is based on essentially instantaneous measurements from many different basins. When relief is considered, regression analysis gives equation (2), N = 0.8147L1.75/R0.25, A = 1, which may also be a growth model. If both equations (1) and (2) are valid as growth models, equation (3), R = 1.899/L, must also be true, when A = 1, for mature basins. Equation (1) is true for long-term changes when erosion can reduce basin relief; equation (2) is true when relief is not reduced because of concomitant uplift. A phase space is defined by the co-ordinates X1 = log vA, X2 = log L, X3 = log R, and X4 = log P. This is a vector four-space over the field of real numbers. A drainage basin can be represented by an image point in the phase space, and θ, F/D2, D, F, etc., are scalar fields, functions of the Xi. The phase space may be an important means of proving an ergodic theorem for drainage systems, which would allow rigorous substitutions of space averages for time averages, as is done in postulating that equations (1) and (2) are growth models. The function relating valley-side slope to the Xi, found by regression analysis, is showing that area and relief have the greatest effect on 0 and channel length is less important. If this can be considered a growth model, θ increases as when R and A are constant. However, in a basin of 1 square mile, because of equation (3), equation (5), θ = 37.94/L0.25, shows that slopes decrease in angle as the drainage net expands, if erosion reduces the relief.

DIMENSIONAL ANALYSIS APPLIED TO FLUVIALLY ERODED LANDFORMS

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 L 2 and length cubed L 3 , 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. Corrsin9s 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.