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95-408: The meanders that form part of either an entrenched river or meander valley are most commonly known as incised meanders . They are commonly classified as either an ingrown meander or an entrenched meander . For a long time, it was argued that ingrown meander occurs when downcutting process is slow and the river can cause lateral erosion , leading to an asymmetric valley . In addition, it

190-427: A region D in three-dimensional space is given by the triple or volume integral of the constant function f ( x , y , z ) = 1 {\displaystyle f(x,y,z)=1} over the region. It is usually written as: ∭ D 1 d x d y d z . {\displaystyle \iiint _{D}1\,dx\,dy\,dz.} In cylindrical coordinates ,

285-437: A reservoir , the container's volume is modeled by shapes and calculated using mathematics. To ease calculations, a unit of volume is equal to the volume occupied by a unit cube (with a side length of one). Because the volume occupies three dimensions, if the metre (m) is chosen as a unit of length, the corresponding unit of volume is the cubic metre (m ). The cubic metre is also a SI derived unit . Therefore, volume has

380-457: A unit dimension of L . The metric units of volume uses metric prefixes , strictly in powers of ten . When applying prefixes to units of volume, which are expressed in units of length cubed, the cube operators are applied to the unit of length including the prefix. An example of converting cubic centimetre to cubic metre is: 2.3 cm = 2.3 (cm) = 2.3 (0.01 m) = 0.0000023 m (five zeros). Commonly used prefixes for cubed length units are

475-612: A formula exists for the shape's boundary. Zero- , one- and two-dimensional objects have no volume; in four and higher dimensions, an analogous concept to the normal volume is the hypervolume. The precision of volume measurements in the ancient period usually ranges between 10–50 mL (0.3–2 US fl oz; 0.4–2 imp fl oz). The earliest evidence of volume calculation came from ancient Egypt and Mesopotamia as mathematical problems, approximating volume of simple shapes such as cuboids , cylinders , frustum and cones . These math problems have been written in

570-452: A maximum at the apex to zero at a crossing point (straight line), also called an inflection, because the curvature changes direction in that vicinity. The radius of the loop is the straight line perpendicular to the down-valley axis intersecting the sinuous axis at the apex. As the loop is not ideal, additional information is needed to characterize it. The orientation angle is the angle between sinuous axis and down-valley axis at any point on

665-405: A meander because helicoidal flow of water keeps the bank washed clean of loose sand, silt, and sediment and subjects it to constant erosion. As a result, the meander erodes and migrates in the direction of the outside bend, forming the cut bank. As the cut bank is undermined by erosion, it commonly collapses as slumps into the river channel. The slumped sediment, having been broken up by slumping,

760-411: A meander is part of an entrenched river or part of a freely meandering river within a floodplain, the term slip-off slope can refer to two different fluvial landforms that comprise the inner, convex, bank of a meander loop. In case of a freely meandering river on a floodplain, a slip-off slope is the inside, gently sloping bank of a meander on which sediments episodically accumulate to form a point bar as

855-431: A meandering watercourse is termed meander geometry or meander planform geometry. It is characterized as an irregular waveform . Ideal waveforms, such as a sine wave , are one line thick, but in the case of a stream the width must be taken into consideration. The bankfull width is the distance across the bed at an average cross-section at the full-stream level, typically estimated by the line of lowest vegetation. As

950-441: A negative value, similar to length and area . Like all continuous monotonic (order-preserving) measures, volumes of bodies can be compared against each other and thus can be ordered. Volume can also be added together and be decomposed indefinitely; the latter property is integral to Cavalieri's principle and to the infinitesimal calculus of three-dimensional bodies. A 'unit' of infinitesimally small volume in integral calculus

1045-421: A non-mathematical utility as well. Streams can be placed in categories arranged by it; for example, when the index is between 1 and 1.5 the river is sinuous, but if between 1.5 and 4, then meandering. The index is a measure also of stream velocity and sediment load, those quantities being maximized at an index of 1 (straight). Volume Volume is a measure of regions in three-dimensional space . It

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1140-428: A river meanders. This type of slip-off slope is located opposite the cutbank. This term can also be applied to the inside, sloping bank of a meandering tidal channel. In case of an entrenched river, a slip-off slope is a gently sloping bedrock surface that rises from the inside, concave bank of an asymmetrically entrenched river. This type of slip-off slope is often covered by a thin, discontinuous layer of alluvium. It

1235-413: A self-intensifying process...in which greater curvature results in more erosion of the bank, which results in greater curvature..." The cross-current along the floor of the channel is part of the secondary flow and sweeps dense eroded material towards the inside of the bend. The cross-current then rises to the surface near the inside and flows towards the outside, forming the helical flow . The greater

1330-419: A series of regular sinuous curves in the channel of a river or other watercourse . It is produced as a watercourse erodes the sediments of an outer, concave bank ( cut bank or river cliff ) and deposits sediments on an inner, convex bank which is typically a point bar . The result of this coupled erosion and sedimentation is the formation of a sinuous course as the channel migrates back and forth across

1425-434: A symmetrical valley sides. He argues that the symmetrical valley sides are the direct result of rapid down-cutting of a watercourse into bedrock. In addition, as proposed by Rich, Thornbury argues that incised valleys with a pronounced asymmetry of cross section, which he called ingrown meanders , are the result of the lateral migration and incision of a meander during a period of slower channel downcutting . Regardless,

1520-413: A waveform the meandering stream follows the down-valley axis, a straight line fitted to the curve such that the sum of all the amplitudes measured from it is zero. This axis represents the overall direction of the stream. At any cross-section the flow is following the sinuous axis, the centerline of the bed. Two consecutive crossing points of sinuous and down-valley axes define a meander loop. The meander

1615-469: Is also very costly when restoration is to be initiated. river entrenchment happens because of the water having the power to cause erosion on the river bed. This increased velocity has shown a negative effect on the riparian habitat , because of increased erosion of the area. For example, a study by Simon revealed that channel incision is a major characteristic of entrenched rivers , and it affects variables in riparian vegetation and growth of fish in

1710-419: Is called lateral accretion. Lateral accretion occurs mostly during high water or floods when the point bar is submerged. Typically, the sediment consists of either sand, gravel, or a combination of both. The sediment comprising some point bars might grade downstream into silty sediments. Because of the decreasing velocity and strength of current from the thalweg of the channel to the upper surface of point bar when

1805-431: Is common for measuring small volume of fluids or granular materials , by using a multiple or fraction of the container. For granular materials, the container is shaken or leveled off to form a roughly flat surface. This method is not the most accurate way to measure volume but is often used to measure cooking ingredients . Air displacement pipette is used in biology and biochemistry to measure volume of fluids at

1900-403: Is often quantified numerically using SI derived units (such as the cubic metre and litre ) or by various imperial or US customary units (such as the gallon , quart , cubic inch ). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that

1995-424: Is produced by the gradual outward migration of the meander as a river cuts downward into bedrock. A terrace on the slip-off slope of a meander spur, known as slip-off slope terrace , can be formed by a brief halt during the irregular incision by an actively meandering river. The meander ratio or sinuosity index is a means of quantifying how much a river or stream meanders (how much its course deviates from

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2090-458: Is readily eroded and carried toward the middle of the channel. The sediment eroded from a cut bank tends to be deposited on the point bar of the next downstream meander, and not on the point bar opposite it. This can be seen in areas where trees grow on the banks of rivers; on the inside of meanders, trees, such as willows, are often far from the bank, whilst on the outside of the bend, the tree roots are often exposed and undercut, eventually leading

2185-536: Is so exceedingly winding that everything winding is called meandering.’ The Meander River is south of Izmir, east of the ancient Greek town of Miletus , now Milet, Turkey. It flows through series of three graben in the Menderes Massif, but has a flood plain much wider than the meander zone in its lower reach. Its modern Turkish name is the Büyük Menderes River . Meanders are a result of

2280-634: Is the volume element ; this formulation is useful when working with different coordinate systems , spaces and manifolds . The oldest way to roughly measure a volume of an object is using the human body, such as using hand size and pinches . However, the human body's variations make it extremely unreliable. A better way to measure volume is to use roughly consistent and durable containers found in nature, such as gourds , sheep or pig stomachs , and bladders . Later on, as metallurgy and glass production improved, small volumes nowadays are usually measured using standardized human-made containers. This method

2375-478: Is the length along the centerline. Once a channel begins to follow a sinusoidal path, the amplitude and concavity of the loops increase dramatically. This is due to the effect of helical flow which sweeps dense eroded material towards the inside of the bend, and leaves the outside of the bend unprotected and vulnerable to accelerated erosion. This establishes a positive feedback loop . In the words of Elizabeth A. Wood: "...this process of making meanders seems to be

2470-405: Is the most common type of fluvial lake, is a crescent-shaped lake that derives its name from its distinctive curved shape. Oxbow lakes are also known as cutoff lakes . Such lakes form regularly in undisturbed floodplains as a result of the normal process of fluvial meandering. Either a river or stream forms a sinuous channel as the outer side of its bends are eroded away and sediments accumulate on

2565-430: Is two consecutive loops pointing in opposite transverse directions. The distance of one meander along the down-valley axis is the meander length or wavelength . The maximum distance from the down-valley axis to the sinuous axis of a loop is the meander width or amplitude . The course at that point is the apex. In contrast to sine waves, the loops of a meandering stream are more nearly circular. The curvature varies from

2660-503: Is used when integrating by an axis parallel to the axis of rotation. The general equation can be written as: V = π ∫ a b | f ( x ) 2 − g ( x ) 2 | d x {\displaystyle V=\pi \int _{a}^{b}\left|f(x)^{2}-g(x)^{2}\right|\,dx} where f ( x ) {\textstyle f(x)} and g ( x ) {\textstyle g(x)} are

2755-542: The Ancient Greeks as Μαίανδρος Maiandros ( Latin : Maeander ), characterised by a very convoluted path along the lower reach. As a result, even in Classical Greece (and in later Greek thought) the name of the river had become a common noun meaning anything convoluted and winding, such as decorative patterns or speech and ideas, as well as the geomorphological feature. Strabo said: ‘...its course

2850-729: The Moscow Mathematical Papyrus (c. 1820 BCE). In the Reisner Papyrus , ancient Egyptians have written concrete units of volume for grain and liquids, as well as a table of length, width, depth, and volume for blocks of material. The Egyptians use their units of length (the cubit , palm , digit ) to devise their units of volume, such as the volume cubit or deny (1 cubit × 1 cubit × 1 cubit), volume palm (1 cubit × 1 cubit × 1 palm), and volume digit (1 cubit × 1 cubit × 1 digit). The last three books of Euclid's Elements , written in around 300 BCE, detailed

2945-483: The Ozark Plateau . As noted above, it was initially either argued or presumed that an incised meander is characteristic of an antecedent stream or river that had incised its channel into underlying strata . An antecedent stream or river is one that maintains its original course and pattern during incision despite the changes in underlying rock topography and rock types. However, later geologists argue that

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3040-449: The bedrock are known as either incised , intrenched , entrenched , inclosed or ingrown meanders . Some Earth scientists recognize and use a finer subdivision of incised meanders. Thornbury argues that incised or inclosed meanders are synonyms that are appropriate to describe any meander incised downward into bedrock and defines enclosed or entrenched meanders as a subtype of incised meanders (inclosed meanders) characterized by

3135-410: The cube , cuboid and cylinder , they have an essentially the same volume calculation formula as one for the prism : the base of the shape multiplied by its height . The calculation of volume is a vital part of integral calculus. One of which is calculating the volume of solids of revolution , by rotating a plane curve around a line on the same plane. The washer or disc integration method

3230-593: The imperial gallon was defined to be the volume occupied by ten pounds of water at 17 °C (62 °F). This definition was further refined until the United Kingdom's Weights and Measures Act 1985 , which makes 1 imperial gallon precisely equal to 4.54609 litres with no use of water. The 1960 redefinition of the metre from the International Prototype Metre to the orange-red emission line of krypton-86 atoms unbounded

3325-522: The sester , amber , coomb , and seam . The sheer quantity of such units motivated British kings to standardize them, culminated in the Assize of Bread and Ale statute in 1258 by Henry III of England . The statute standardized weight, length and volume as well as introduced the peny, ounce, pound, gallon and bushel. In 1618, the London Pharmacopoeia (medicine compound catalog) adopted

3420-439: The volume integral is ∭ D r d r d θ d z , {\displaystyle \iiint _{D}r\,dr\,d\theta \,dz,} In spherical coordinates (using the convention for angles with θ {\displaystyle \theta } as the azimuth and φ {\displaystyle \varphi } measured from the polar axis; see more on conventions ),

3515-543: The Middle Reach pit, which in turn led to the creation of the entrenched river. According to Posamentier, in the 1950s and 1980s, the Russian River had gravel pits and dry creek . However, over time the river has become entrenched due to mining (p. 1777). Urbanization and clearing of vegetation increase runoff water, which in turn increases water volume , especially during rainy seasons. Therefore,

3610-491: The Roman gallon or congius as a basic unit of volume and gave a conversion table to the apothecaries' units of weight. Around this time, volume measurements are becoming more precise and the uncertainty is narrowed to between 1–5 mL (0.03–0.2 US fl oz; 0.04–0.2 imp fl oz). Around the early 17th century, Bonaventura Cavalieri applied the philosophy of modern integral calculus to calculate

3705-435: The absence of secondary flow we would expect low fluid velocity at the outside bend and high fluid velocity at the inside bend. This classic fluid mechanics result is irrotational vortex flow. In the context of meandering rivers, its effects are dominated by those of secondary flow. Secondary flow : A force balance exists between pressure forces pointing to the inside bend of the river and centrifugal forces pointing to

3800-409: The area or when the lowering of the sea level occurs. It can also be caused by increased level of downcutting or a collapse of moraine -dammed lake downstream, or capture of the river by another river. Moreover, the process of river rejuvenation can also be the cause of river entrenchment , especially when the process has occurred due to tectonic uplift . River rejuvenation increases the power of

3895-716: The areas (p. 528). Lowering of the channels means that the ground water level has also reduced. Specifically, the development of the entrenched river reduces the amount of ground water due to water loss through infiltration . The movement of the base level changes tributaries and the entrenchment of a tributary channel. Studies of various rivers have revealed that the process of river entrenchment has been associated with adjustment of river positions through bank erosion , as well as widening. River entrenchment causes negative impacts, such as accelerated stream bank erosion , land loss, loss of aquatic habitat , loss of land productivity, lowering of water table and sedimentation of

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3990-411: The average fullbank channel width. The length of the stream is measured by channel, or thalweg, length over the reach, while the bottom value of the ratio is the downvalley length or air distance of the stream between two points on it defining the reach. The sinuosity index plays a part in mathematical descriptions of streams. The index may require elaboration, because the valley may meander as well—i.e.,

4085-422: The axis of a floodplain . The zone within which a meandering stream periodically shifts its channel is known as a meander belt . It typically ranges from 15 to 18 times the width of the channel. Over time, meanders migrate downstream, sometimes in such a short time as to create civil engineering challenges for local municipalities attempting to maintain stable roads and bridges. The degree of meandering of

4180-462: The base of the river and they occur when the river base level has reduced, thus giving the river enough power for vertical erosion to take place. Some studies also indicate that anthropogenic factors such as clearing of vegetation , development of dams , and reservoir and urbanization are also causes of river entrenchment. For example, gravel mining along the Russian River developed

4275-554: The bottom from the outside to the inside. The flow is supplied by a counter-flow across the surface from the inside to the outside. This entire situation is very similar to the Tea leaf paradox . This secondary flow carries sediment from the outside of the bend to the inside making the river more meandering. As to why streams of any size become sinuous in the first place, there are a number of theories, not necessarily mutually exclusive. The stochastic theory can take many forms but one of

4370-429: The boundary layer. Therefore, within the boundary layer, pressure force dominates and fluid moves along the bottom of the river from the outside bend to the inside bend. This initiates helicoidal flow: Along the river bed, fluid roughly follows the curve of the channel but is also forced toward the inside bend; away from the river bed, fluid also roughly follows the curve of the channel but is forced, to some extent, from

4465-559: The case of the Anderson Bottom Rincon, incised meanders that have either steep-sided, often vertical walls, are often, but not always, known as rincons in the southwest United States . Rincon in English is a nontechnical word in the southwest United States for either a small secluded valley, an alcove or angular recess in a cliff, or a bend in a river. The meanders of a stream or river that has cut its bed down into

4560-470: The channel of a river, stream, or other watercourse is measured by its sinuosity . The sinuosity of a watercourse is the ratio of the length of the channel to the straight line down-valley distance. Streams or rivers with a single channel and sinuosities of 1.5 or more are defined as meandering streams or rivers. The term derives from the winding river Menderes located in Asia-Minor and known to

4655-408: The consequences of incised meander are associated with accelerated stream bank erosion , land loss , aquatic habitat loss, as well as lowering the water table. Additionally, the study also found that incised meanders also cause loss of land productivity and downstream sedimentation (p. 2). The factors are likely to affect not only the economic development on the land where it passes but it

4750-487: The contained volume does not need to fill towards the container's capacity, or vice versa. Containers can only hold a specific amount of physical volume, not weight (excluding practical concerns). For example, a 50,000 bbl (7,900,000 L) tank that can just hold 7,200 t (15,900,000 lb) of fuel oil will not be able to contain the same 7,200 t (15,900,000 lb) of naphtha , due to naphtha's lower density and thus larger volume. For many shapes such as

4845-524: The container could hold, rather than the amount of space the container itself displaces. By metonymy , the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume ). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas . Volumes of more complicated shapes can be calculated with integral calculus if

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4940-399: The cubic millimetre (mm ), cubic centimetre (cm ), cubic decimetre (dm ), cubic metre (m ) and the cubic kilometre (km ). The conversion between the prefix units are as follows: 1000 mm = 1 cm , 1000 cm = 1 dm , and 1000 dm = 1 m . The metric system also includes the litre (L) as a unit of volume, where 1 L = 1 dm = 1000 cm = 0.001 m . For

5035-408: The curvature of the bend, and the faster the flow, the stronger is the cross-current and the sweeping. Due to the conservation of angular momentum the speed on the inside of the bend is faster than on the outside. Since the flow velocity is diminished, so is the centrifugal pressure. The pressure of the super-elevated column prevails, developing an unbalanced gradient that moves water back across

5130-405: The downvalley length is not identical to the reach. In that case the valley index is the meander ratio of the valley while the channel index is the meander ratio of the channel. The channel sinuosity index is the channel length divided by the valley length and the standard sinuosity index is the channel index divided by the valley index. Distinctions may become even more subtle. Sinuosity Index has

5225-533: The exact formulas for calculating the volume of parallelepipeds , cones, pyramids , cylinders, and spheres . The formula were determined by prior mathematicians by using a primitive form of integration , by breaking the shapes into smaller and simpler pieces. A century later, Archimedes ( c.  287 – 212 BCE ) devised approximate volume formula of several shapes using the method of exhaustion approach, meaning to derive solutions from previous known formulas from similar shapes. Primitive integration of shapes

5320-402: The flowing water and, therefore, the process of erosion is accelerated. Studies show that tectonic movement , particularly movement associated with uplift , can influence spatial patterns of erosion and sedimentation . Although it is very difficult to offer detailed information of past tectonic activity, the basic temporal and spatial scale can show evidence of how this movement leads to

5415-529: The formation of an entrenched river. Various authors have used an entrenched river as evidence of tectonic movement in the past, and in this way they have proven the significant role of tectonic uplifts in the formation of an entrenched river. Several studies have cited incised meander a major feature of river entrenchment, as an impact of river rejuvenation. On the other hand, scholars argue that incised meanders and entrenched meanders are features formed before river rejuvenation. Incised meanders occur at

5510-565: The formation of both entrenched meanders and ingrown meanders is thought to require that base level falls as a result of either relative change in mean sea level , isostatic or tectonic uplift, the breach of an ice or landslide dam, or regional tilting. Classic examples of incised meanders are associated with rivers in the Colorado Plateau , the Kentucky River Palisades in central Kentucky , and streams in

5605-428: The full force of the flood. After a cutoff meander is formed, river water flows into its end from the river builds small delta-like feature into either end of it during floods. These delta-like features block either end of the cutoff meander to form a stagnant oxbow lake that is separated from the flow of the fluvial channel and independent of the river. During floods, the flood waters deposit fine-grained sediment into

5700-404: The fullbank channel width and 3 to 5 times, with an average of 4.7 times, the radius of curvature at the apex. This radius is 2–3 times the channel width. A meander has a depth pattern as well. The cross-overs are marked by riffles , or shallow beds, while at the apices are pools. In a pool direction of flow is downward, scouring the bed material. The major volume, however, flows more slowly on

5795-508: The globe involving river restoration have been conducted, and a good example of such projects is Maggie Creek, Nevada . The project was completed in 1990 on upper Maggie Creek in Nevada . It was a partnership between the government and a private ranch, and the project entailed straightening of many miles of unstable gravel bed regarded as the C4/D4 type. Meanders A meander is one of

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5890-570: The golden crown to find its volume, and thus its density and purity, due to the extreme precision involved. Instead, he likely have devised a primitive form of a hydrostatic balance . Here, the crown and a chunk of pure gold with a similar weight are put on both ends of a weighing scale submerged underwater, which will tip accordingly due to the Archimedes' principle . In the Middle Ages , many units for measuring volume were made, such as

5985-462: The increased level of vertical erosion of the river increases the power of the water, leading to erosion of the river. A study of San Pedro River and another river in the southwest have indicated that floods were the main cause of river entrenchment in the 18th century. The study shows that increased population and human activities in these places increased floods and, consequently, the volume of runoff water (Hereford 43). Rosgen indicates that

6080-416: The inner side, which forms a meandering horseshoe-shaped bend. Eventually as the result of its meandering, the fluvial channel cuts through the narrow neck of the meander and forms a cutoff meander. The final break-through of the neck, which is called a neck cutoff , often occurs during a major flood because that is when the watercourse is out of its banks and can flow directly across the neck and erode it with

6175-443: The inside bank of a river bend. On the inside bend, this sediment and debris is eventually deposited on the slip-off slope of a point bar. Scroll-bars are a result of continuous lateral migration of a meander loop that creates an asymmetrical ridge and swale topography on the inside of the bends. The topography is generally parallel to the meander, and is related to migrating bar forms and back bar chutes, which carve sediment from

6270-406: The inside of the bend where, due to decreased velocity, it deposits sediment. The line of maximum depth, or channel, is the thalweg or thalweg line. It is typically designated the borderline when rivers are used as political borders. The thalweg hugs the outer banks and returns to center over the riffles. The meander arc length is the distance along the thalweg over one meander. The river length

6365-450: The inside to the outside bend. The higher velocities at the outside bend lead to higher shear stresses and therefore result in erosion. Similarly, lower velocities at the inside bend cause lower shear stresses and deposition occurs. Thus meander bends erode at the outside bend, causing the river to becoming increasingly sinuous (until cutoff events occur). Deposition at the inside bend occurs such that for most natural meandering rivers,

6460-404: The interaction of water flowing through a curved channel with the underlying river bed. This produces helicoidal flow , in which water moves from the outer to the inner bank along the river bed, then flows back to the outer bank near the surface of the river. This in turn increases carrying capacity for sediments on the outer bank and reduces it on the inner bank, so that sediments are eroded from

6555-415: The litre unit, the commonly used prefixes are the millilitre (mL), centilitre (cL), and the litre (L), with 1000 mL = 1 L, 10 mL = 1 cL, 10 cL = 1 dL, and 10 dL = 1 L. Various other imperial or U.S. customary units of volume are also in use, including: Capacity is the maximum amount of material that a container can hold, measured in volume or weight . However,

6650-420: The meanders are fixed. Various mathematical formulae relate the variables of the meander geometry. As it turns out some numerical parameters can be established, which appear in the formulae. The waveform depends ultimately on the characteristics of the flow but the parameters are independent of it and apparently are caused by geologic factors. In general the meander length is 10–14 times, with an average 11 times,

6745-597: The metre, cubic metre, and litre from physical objects. This also make the metre and metre-derived units of volume resilient to changes to the International Prototype Metre. The definition of the metre was redefined again in 1983 to use the speed of light and second (which is derived from the caesium standard ) and reworded for clarity in 2019 . As a measure of the Euclidean three-dimensional space , volume cannot be physically measured as

6840-598: The microscopic scale. Calibrated measuring cups and spoons are adequate for cooking and daily life applications, however, they are not precise enough for laboratories . There, volume of liquids is measured using graduated cylinders , pipettes and volumetric flasks . The largest of such calibrated containers are petroleum storage tanks , some can hold up to 1,000,000  bbl (160,000,000 L) of fluids. Even at this scale, by knowing petroleum's density and temperature, very precise volume measurement in these tanks can still be made. For even larger volumes such as in

6935-509: The modern integral calculus, which remains in use in the 21st century. On 7 April 1795, the metric system was formally defined in French law using six units. Three of these are related to volume: the stère  (1 m ) for volume of firewood; the litre  (1 dm ) for volumes of liquid; and the gramme , for mass—defined as the mass of one cubic centimetre of water at the temperature of melting ice. Thirty years later in 1824,

7030-418: The more heterogeneous braided river deposits. There are two distinct patterns of scroll-bar depositions; the eddy accretion scroll bar pattern and the point-bar scroll pattern. When looking down the river valley they can be distinguished because the point-bar scroll patterns are convex and the eddy accretion scroll bar patterns are concave. Scroll bars often look lighter at the tops of the ridges and darker in

7125-504: The most general statements is that of Scheidegger: "The meander train is assumed to be the result of the stochastic fluctuations of the direction of flow due to the random presence of direction-changing obstacles in the river path." Given a flat, smooth, tilted artificial surface, rainfall runs off it in sheets, but even in that case adhesion of water to the surface and cohesion of drops produce rivulets at random. Natural surfaces are rough and erodible to different degrees. The result of all

7220-514: The outer bank and redeposited on the inner bank of the next downstream meander. When a fluid is introduced to an initially straight channel which then bends, the sidewalls induce a pressure gradient that causes the fluid to alter course and follow the bend. From here, two opposing processes occur: (1) irrotational flow and (2) secondary flow . For a river to meander, secondary flow must dominate. Irrotational flow : From Bernoulli's equations, high pressure results in low velocity. Therefore, in

7315-403: The outside bend of the river. In the context of meandering rivers, a boundary layer exists within the thin layer of fluid that interacts with the river bed. Inside that layer and following standard boundary-layer theory, the velocity of the fluid is effectively zero. Centrifugal force, which depends on velocity, is also therefore effectively zero. Pressure force, however, remains unaffected by

7410-529: The outside of the curve and deposit sediment in the slower flowing water on the inside of the loop, in a process called lateral accretion. Scroll-bar sediments are characterized by cross-bedding and a pattern of fining upward. These characteristics are a result of the dynamic river system, where larger grains are transported during high energy flood events and then gradually die down, depositing smaller material with time (Batty 2006). Deposits for meandering rivers are generally homogeneous and laterally extensive unlike

7505-415: The oxbow lake. As a result, oxbow lakes tend to become filled in with fine-grained, organic-rich sediments over time. A point bar , which is also known as a meander bar , is a fluvial bar that is formed by the slow, often episodic, addition of individual accretions of noncohesive sediment on the inside bank of a meander by the accompanying migration of the channel toward its outer bank. This process

7600-475: The physical factors acting at random is channels that are not straight, which then progressively become sinuous. Even channels that appear straight have a sinuous thalweg that leads eventually to a sinuous channel. In the equilibrium theory, meanders decrease the stream gradient until an equilibrium between the erodibility of the terrain and the transport capacity of the stream is reached. A mass of water descending must give up potential energy , which, given

7695-409: The plane curve boundaries. The shell integration method is used when integrating by an axis perpendicular to the axis of rotation. The equation can be written as: V = 2 π ∫ a b x | f ( x ) − g ( x ) | d x {\displaystyle V=2\pi \int _{a}^{b}x|f(x)-g(x)|\,dx} The volume of

7790-474: The river downstream. However, in order to offset these problems, channel restoration measures focusing on restoration of the river in its original or to its previous characteristics are available. Though to achieve good results, a good understanding of the river patterns and profile of stable channels is a critical requirement. The process also needs an elaborate procedure to be followed to ensure all important factors and actions are followed. Numerous projects across

7885-691: The river width remains nearly constant, even as the river evolves. In a speech before the Prussian Academy of Sciences in 1926, Albert Einstein suggested that because the Coriolis force of the earth can cause a small imbalance in velocity distribution, such that velocity on one bank is higher than on the other, it could trigger the erosion on one bank and deposition of sediment on the other that produces meanders However, Coriolis forces are likely insignificant compared with other forces acting to produce river meanders. The technical description of

7980-416: The same velocity at the end of the drop as at the beginning, is removed by interaction with the material of the stream bed. The shortest distance; that is, a straight channel, results in the highest energy per unit of length, disrupting the banks more, creating more sediment and aggrading the stream. The presence of meanders allows the stream to adjust the length to an equilibrium energy per unit length in which

8075-417: The sediment is deposited the vertical sequence of sediments comprising a point bar becomes finer upward within an individual point bar. For example, it is typical for point bars to fine upward from gravel at the base to fine sands at the top. The source of the sediment is typically upstream cut banks from which sand, rocks and debris has been eroded, swept, and rolled across the bed of the river and downstream to

8170-505: The shape of an incised meander is not always, if ever, "inherited", e.g., strictly from an antecedent meandering stream where its meander pattern could freely develop on a level floodplain. Instead, they argue that as fluvial incision of bedrock proceeds, the stream course is significantly modified by variations in rock type and fractures , faults , and other geological structures into either lithologically conditioned meanders or structurally controlled meanders . The oxbow lake , which

8265-445: The shortest possible path). It is calculated as the length of the stream divided by the length of the valley . A perfectly straight river would have a meander ratio of 1 (it would be the same length as its valley), while the higher this ratio is above 1, the more the river meanders. Sinuosity indices are calculated from the map or from an aerial photograph measured over a distance called the reach , which should be at least 20 times

8360-411: The sinuous axis. A loop at the apex has an outer or concave bank and an inner or convex bank. The meander belt is defined by an average meander width measured from outer bank to outer bank instead of from centerline to centerline. If there is a flood plain , it extends beyond the meander belt. The meander is then said to be free—it can be found anywhere in the flood plain. If there is no flood plain,

8455-419: The stream carries away all the sediment that it produces. Geomorphic refers to the surface structure of the terrain. Morphotectonic means having to do with the deeper, or tectonic (plate) structure of the rock. The features included under these categories are not random and guide streams into non-random paths. They are predictable obstacles that instigate meander formation by deflecting the stream. For example,

8550-403: The stream might be guided into a fault line (morphotectonic). A cut bank is an often vertical bank or cliff that forms where the outside, concave bank of a meander cuts into the floodplain or valley wall of a river or stream. A cutbank is also known either as a river-cut cliff , river cliff , or a bluff and spelled as cutbank . Erosion that forms a cut bank occurs at the outside bank of

8645-416: The swales. This is because the tops can be shaped by wind, either adding fine grains or by keeping the area unvegetated, while the darkness in the swales can be attributed to silts and clays washing in during high water periods. This added sediment in addition to water that catches in the swales is in turn is a favorable environment for vegetation that will also accumulate in the swales. Depending upon whether

8740-406: The trees to fall into the river. A meander cutoff , also known as either a cutoff meander or abandoned meander , is a meander that has been abandoned by its stream after the formation of a neck cutoff. A lake that occupies a cutoff meander is known as an oxbow lake . Cutoff meanders that have cut downward into the underlying bedrock are known in general as incised cutoff meanders . As in

8835-399: The volume of any object. He devised Cavalieri's principle , which said that using thinner and thinner slices of the shape would make the resulting volume more and more accurate. This idea would then be later expanded by Pierre de Fermat , John Wallis , Isaac Barrow , James Gregory , Isaac Newton , Gottfried Wilhelm Leibniz and Maria Gaetana Agnesi in the 17th and 18th centuries to form

8930-549: Was also argued for a long time that an entrenched meander forms when there is a rapid incision of the river bed such that the river does not have the opportunity to erode the lateral side. This leads to symmetrical valleys with a gorge -like appearance. However, more detailed studies have shown the development of ingrown meanders versus entrenched meanders depends on a complex mixture of factors such as bedrock lithology, tectonic activity, and climate. As observed above, an entrenched river can be caused by either tectonic uplift in

9025-501: Was also discovered independently by Liu Hui in the 3rd century CE, Zu Chongzhi in the 5th century CE, the Middle East and India . Archimedes also devised a way to calculate the volume of an irregular object, by submerging it underwater and measure the difference between the initial and final water volume. The water volume difference is the volume of the object. Though highly popularized, Archimedes probably does not submerge

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