Shallow water equations - Knowledge (XXG)

6153: 1248: 661: 4874: 602: 4456: 1243:{\displaystyle {\begin{aligned}{\frac {\partial h}{\partial t}}&+{\frac {\partial }{\partial x}}{\Bigl (}(H+h)u{\Bigr )}+{\frac {\partial }{\partial y}}{\Bigl (}(H+h)v{\Bigr )}=0,\\{\frac {\partial u}{\partial t}}&+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}-fv=-g{\frac {\partial h}{\partial x}}-ku+\nu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right),\\{\frac {\partial v}{\partial t}}&+u{\frac {\partial v}{\partial x}}+v{\frac {\partial v}{\partial y}}+fu=-g{\frac {\partial h}{\partial y}}-kv+\nu \left({\frac {\partial ^{2}v}{\partial x^{2}}}+{\frac {\partial ^{2}v}{\partial y^{2}}}\right),\end{aligned}}} 153: 4869:{\displaystyle {\begin{aligned}&\rho B{\frac {\partial \zeta }{\partial t}}+{\frac {\partial }{\partial x}}\left({\frac {\partial H}{\partial u}}\right)=\rho \left(B{\frac {\partial \zeta }{\partial t}}+{\frac {\partial (Au)}{\partial x}}\right)=\rho \left({\frac {\partial A}{\partial t}}+{\frac {\partial (Au)}{\partial x}}\right)=0,\\&\rho B{\frac {\partial u}{\partial t}}+{\frac {\partial }{\partial x}}\left({\frac {\partial H}{\partial \zeta }}\right)=\rho B\left({\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+g{\frac {\partial \zeta }{\partial x}}\right)=0,\end{aligned}}} 597:{\displaystyle {\begin{aligned}{\frac {\partial (\rho \eta )}{\partial t}}&+{\frac {\partial (\rho \eta u)}{\partial x}}+{\frac {\partial (\rho \eta v)}{\partial y}}=0,\\{\frac {\partial (\rho \eta u)}{\partial t}}&+{\frac {\partial }{\partial x}}\left(\rho \eta u^{2}+{\frac {1}{2}}\rho g\eta ^{2}\right)+{\frac {\partial (\rho \eta uv)}{\partial y}}=0,\\{\frac {\partial (\rho \eta v)}{\partial t}}&+{\frac {\partial }{\partial y}}\left(\rho \eta v^{2}+{\frac {1}{2}}\rho g\eta ^{2}\right)+{\frac {\partial (\rho \eta uv)}{\partial x}}=0.\end{aligned}}} 4306: 3538: 6678: 4007: 5472: 1881: 3659: 1808: 8097: 3192: 5186: 8985: 4301:{\displaystyle {\begin{aligned}&{\frac {\mathrm {d} }{\mathrm {d} t}}\left(u+2{\sqrt {gh}}\right)=g\left(S-S_{f}\right)&&{\text{along}}\quad {\frac {\mathrm {d} x}{\mathrm {d} t}}=u+{\sqrt {gh}}\quad {\text{and}}\\&{\frac {\mathrm {d} }{\mathrm {d} t}}\left(u-2{\sqrt {gh}}\right)=g\left(S-S_{f}\right)&&{\text{along}}\quad {\frac {\mathrm {d} x}{\mathrm {d} t}}=u-{\sqrt {gh}}.\end{aligned}}} 1540: 9006: 106: 5107: 3533:{\displaystyle {\begin{aligned}A(\sigma ,x)&=\int _{0}^{\sigma }B(\sigma ',x)\;\mathrm {d} \sigma ',\\I_{1}(\sigma ,x)&=\int _{0}^{\sigma }(\sigma -\sigma ')\,B(\sigma ^{\prime },x)\;\mathrm {d} \sigma '\qquad {\text{and}}\\I_{2}(\sigma ,x)&=\int _{0}^{\sigma }(\sigma -\sigma ')\,{\frac {\partial B(\sigma ',x)}{\partial x}}\;\mathrm {d} \sigma '.\end{aligned}}} 8086: 8995: 5467:{\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+v{\frac {\partial u}{\partial y}}+w{\frac {\partial u}{\partial z}}=-{\frac {\partial p}{\partial x}}{\frac {1}{\rho }}+\nu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\right)+f_{x},} 74:, in the case where the horizontal length scale is much greater than the vertical length scale. Under this condition, conservation of mass implies that the vertical velocity scale of the fluid is small compared to the horizontal velocity scale. It can be shown from the momentum equation that vertical pressure gradients are nearly 1803:{\displaystyle {\begin{aligned}{\frac {\partial h}{\partial t}}&+H\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0,\\{\frac {\partial u}{\partial t}}&-fv=-g{\frac {\partial h}{\partial x}}-ku,\\{\frac {\partial v}{\partial t}}&+fu=-g{\frac {\partial h}{\partial y}}-kv.\end{aligned}}} 78:, and that horizontal pressure gradients are due to the displacement of the pressure surface, implying that the horizontal velocity field is constant throughout the depth of the fluid. Vertically integrating allows the vertical velocity to be removed from the equations. The shallow-water equations are thus derived. 5982: 2128: 1486: 3126: 96:

Shallow-water equation models have only one vertical level, so they cannot directly encompass any factor that varies with height. However, in cases where the mean state is sufficiently simple, the vertical variations can be separated from the horizontal and several sets of shallow-water equations can

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While a vertical velocity term is not present in the shallow-water equations, note that this velocity is not necessarily zero. This is an important distinction because, for example, the vertical velocity cannot be zero when the floor changes depth, and thus if it were zero only flat floors would be

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The pressure gradient term (c) describes how pressure changes with position, and since the pressure is assumed hydrostatic, this is the change in head over position. The friction term (d) accounts for losses in energy due to friction, while the gravity term (e) is the acceleration due to bed slope.

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Animation of the linearized shallow-water equations for a rectangular basin, without friction and Coriolis force. The water experiences a splash which generates surface gravity waves that propagate away from the splash location and reflect off the basin walls. The animation is created using the

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The local acceleration (a) can also be thought of as the "unsteady term" as this describes some change in velocity over time. The convective acceleration (b) is an acceleration caused by some change in velocity over position, for example the speeding up or slowing down of a fluid entering a

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effects. Shallow-water equations are especially suitable to model tides which have very large length scales (over hundred of kilometers). For tidal motion, even a very deep ocean may be considered as shallow as its depth will always be much smaller than the tidal wavelength.

6500: 5853: 31: 1833:. They can be viewed as a contraction of the two-dimensional (2-D) shallow-water equations, which are also known as the two-dimensional Saint-Venant equations. The 1-D Saint-Venant equations contain to a certain extent the main characteristics of the channel 5026:

The diffusive wave is valid when the inertial acceleration is much smaller than all other forms of acceleration, or in other words when there is primarily subcritical flow, with low Froude values. Models that use the diffusive wave assumption include

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The dynamic wave is the full one-dimensional Saint-Venant equation. It is numerically challenging to solve, but is valid for all channel flow scenarios. The dynamic wave is used for modeling transient storms in modeling programs including

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of the phenomenon they are supposed to model has to be much larger than the depth of the basin where the phenomenon takes place. Somewhat smaller wavelengths can be handled by extending the shallow-water equations using the

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The kinematic wave is valid when the change in wave height over distance and velocity over distance and time is negligible relative to the bed slope, e.g. for shallow flows over steep slopes. The kinematic wave is used in

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For the diffusive wave it is assumed that the inertial terms are less than the gravity, friction, and pressure terms. The diffusive wave can therefore be more accurately described as a non-inertia wave, and is written as:

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In the order of increasing simplifications, by removing some terms of the full 1D Saint-Venant equations (aka Dynamic wave equation), we get the also classical Diffusive wave equation and Kinematic wave equation.

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Output from a shallow-water equation model of water in a bathtub. The water experiences five splashes which generate surface gravity waves that propagate away from the splash locations and reflect off the bathtub

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usable with the shallow-water equations. Once a solution (i.e. the horizontal velocities and free surface displacement) has been found, the vertical velocity can be recovered via the continuity equation.

6152: 654:, the non-conservative form of the shallow-water equations is obtained. Since velocities are not subject to a fundamental conservation equation, the non-conservative forms do not hold across a shock or 4313:

The characteristics and Riemann invariants provide important information on the behavior of the flow, as well as that they may be used in the process of obtaining (analytical or numerical) solutions.

2722: 6713:. Some alternate models have been proposed which prevent shock formation. One alternative is to modify the "pressure term" in the momentum equation, but it results in a complicated expression for 85:

Situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are common, so the shallow-water equations are widely applicable. They are used with

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it is assumed that the flow is uniform, and that the friction slope is approximately equal to the slope of the channel. This simplifies the full Saint-Venant equation to the kinematic wave:

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Anderson, Dalton; Harris, Matthew; Hartle, Harrison; Nicolsky, Dmitry; Pelinovsky, Efim; Raz, Amir; Rybkin, Alexei (2017-02-02). "Run-Up of Long Waves in Piecewise Sloping U-Shaped Bays".

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Harris, M. W.; Nicolsky, D. J.; Pelinovsky, E. N.; Rybkin, A. V. (2015-03-01). "Runup of Nonlinear Long Waves in Trapezoidal Bays: 1-D Analytical Theory and 2-D Numerical Computations".

3824: 1912: 6709:. However, there are also some disadvantages as far as geophysical applications are concerned - it has a non-quadratic expression for total energy and a tendency for waves to become 4002: 1496: 6129: 6049: 3950: 2758: 1872:

along rivers (including evaluation of measures to reduce the risks of flooding), dam break analysis, storm pulses in an open channel, as well as storm runoff in overland flow.

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A numerical model simulating water flow and contaminant and sediment transport in watershed systems of 1-D stream-river network, 2-D overland regime, and 3-D subsurface media

6368: 5977:{\displaystyle -{\frac {\partial p}{\partial x}}{\frac {1}{\rho }}=-{\frac {1}{\rho }}{\frac {\rho g\left(\partial h\right)}{\partial x}}=-g{\frac {\partial h}{\partial x}}.} 6650: 6594:

where (a) is the local acceleration term, (b) is the convective acceleration term, (c) is the pressure gradient term, (d) is the friction term, and (e) is the gravity term.

5696: 6314: 8725: 5084: 5797: 2123:{\displaystyle {\frac {\partial u}{\partial t}}+u\,{\frac {\partial u}{\partial x}}+g\,{\frac {\partial \zeta }{\partial x}}=-{\frac {P}{A}}\,{\frac {\tau }{\rho }},} 8715: 7774: 6910: 1822: 60: 5541: 6208: 6504: 6318:

Assuming the energy grade line is not the same as the channel slope, and for a reach of consistent slope there is a consistent friction loss, it follows that:

3121:{\displaystyle {\frac {\partial Q}{\partial t}}+{\frac {\partial }{\partial x}}\left({\frac {Q^{2}}{A}}+g\,I_{1}\right)+g\,A\,\left(S_{f}-S\right)-g\,I_{2}=0,} 2616: 5680:{\displaystyle \nu \left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}+{\frac {\partial ^{2}u}{\partial z^{2}}}\right)=0.} 4310:

The Riemann invariants and method of characteristics for a prismatic channel of arbitrary cross-section are described by Didenkulova & Pelinovsky (2011).

2898:{\displaystyle {\frac {\partial u}{\partial t}}+u\,{\frac {\partial u}{\partial x}}+g\,{\frac {\partial h}{\partial x}}+g\,\left(S_{\mathrm {f} }-S\right)=0.} 7400:

Scharffenberg, W. A., and M. J. Fleming (2006), Hydrologic Modeling System HEC-HMS: User's Manual, US Army Corps of Engineers, Hydrologic Engineering Center.

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Havnø, K., M. Madsen, J. Dørge, and V. Singh (1995), MIKE 11-a generalized river modelling package, Computer models of watershed hydrology., 733–782.

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because they are significantly easier to solve than the full shallow-water equations. Common applications of the 1-D Saint-Venant equations include

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In applications, depending on the problem at hand, there often is a preference for using either the momentum equation in non-conservation form, (

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Bates, P., T. Fewtrell, M. Trigg, and J. Neal (2008), LISFLOOD-FP user manual and technical note, code release 4.3. 6, University of Bristol.

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Searby, D.; Dean, A.; Margetts J. (1998), Christchurch harbour Hydroworks modelling., Proceedings of the WAPUG Autumn meeting, Blackpool, UK.

7058: 7022: 6788: 6772: 2252: 6970: 2380: 51:(or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a 9031: 8278: 8168: 7767: 8873: 8300: 8188: 7726: 7479: 8720: 7991: 7422: 7310: 6999: 5802: 5150: 8138: 5117: 6495:{\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+g{\frac {\partial h}{\partial x}}+g(S_{f}-S)=0,} 8178: 5035:

software this options is also available, since the 2 inertia terms (or any of them) can be removed in option from the interface.

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generation and propagation, as computed with the shallow-water equations (red line; without frequency dispersion)), and with a

658:. Also included are the appropriate terms for Coriolis, frictional and viscous forces, to obtain (for constant fluid density): 8581: 6649: 9041: 8988: 7760: 3955: 2271:) is obtained from the geometry of cross sections – by providing a functional relationship between the cross-sectional area 6083: 5989: 3906: 7034: 1889: 8036: 4922: 7746: 6664:

with an oscillatory tail staying behind. The shallow-water equations (red line) form a steep front, which will lead to

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Brunner, G. W. (1995), HEC-RAS River Analysis System. Hydraulic Reference Manual. Version 1.0 Rep., DTIC Document.

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Didenkulova, I.; Pelinovsky, E. (2011). "Rogue waves in nonlinear hyperbolic systems (shallow-water framework)".

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Novak, P., et al., Hydraulic Modelling – An Introduction: Principles, Methods and Applications. 2010: CRC Press.

7170:"Run-up of nonlinear long waves in U-shaped bays of finite length: analytical theory and numerical computations" 5050: 8898: 8273: 8263: 8203: 7839: 7809: 6642: 5770: 1901: 1834: 123: 6677: 8935: 8918: 8755: 8248: 8113: 8041: 7934: 6734: 6717:. Another option is to modify the non-linear terms in all equations, which gives a quadratic expression for 6885:

Joint Environment Agency/Defra Flood and Coastal Erosion Risk Management Research and Development Programme

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in a smaller domain (e.g. surface waves in a bath). In order for shallow-water equations to be valid, the

7446: 8763: 8745: 8253: 8148: 7783: 4326: 2949: 4430:{\displaystyle H=\rho \int \left({\frac {1}{2}}Au^{2}+{\frac {1}{2}}gB\zeta ^{2}\right)\mathrm {d} x,} 17: 9046: 8950: 8783: 8486: 8343: 8208: 7919: 7650: 7601: 7544: 7259: 7126: 7083: 1431: 1286: 1268: 119: 75: 642:. The first equation is derived from mass conservation, the second two from momentum conservation. 8945: 8830: 8825: 8551: 8223: 8183: 7899: 7046: 6722: 6706: 6660:(blue line; with frequency dispersion). Observe that the Boussinesq-type model (blue line) forms a 4907: 2559: 1893: 1869: 1849: 1520: 1477: 90: 6795: 8888: 8601: 8591: 8556: 8456: 8441: 8338: 7570: 7283: 7217:

Garayshin, V. V.; Harris, M. W.; Nicolsky, D. J.; Pelinovsky, E. N.; Rybkin, A. V. (2016-04-10).

7199: 7150: 7099: 7050: 3857: 3846: 1981:{\displaystyle {\frac {\partial A}{\partial t}}+{\frac {\partial \left(Au\right)}{\partial x}}=0} 7373:

DHI (Danish Hydraulic Institute) (2011), MIKE SHE User Manual Volume 2: Reference Guide, edited.

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Set of partial differential equations that describe the flow below a pressure surface in a fluid

3658: 2562:, expressing conservation of water volume for this incompressible homogeneous fluid. Equation ( 8970: 8960: 8903: 8883: 8566: 8531: 8466: 8446: 8436: 8318: 8006: 7864: 7722: 7704: 7666: 7619: 7562: 7512: 7485: 7475: 7428: 7418: 7316: 7306: 7275: 7191: 7168:

Harris, M. W.; Nicolsky, D. J.; Pelinovsky, E. N.; Pender, J. M.; Rybkin, A. V. (2016-05-01).

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All of these assumptions combined arrives at the 1-dimensional Saint-Venant equation in the

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Assuming also that the pressure distribution is approximately hydrostatic it follows that:

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Characteristics, domain of dependence and region of influence, associated with location

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is small. Assuming also that the wave height is very small compared to the mean height (

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Shallow-water equations, in its non-linear form, is an obvious candidate for modelling

5044: 3643: 3602: 2360: 1904:– as derived and posed by Saint-Venant in his 1871 paper (equations 19 & 20) – is: 1841: 1830: 1500: 1435: 1309:

is the height deviation of the horizontal pressure surface from its mean height, where

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A snapshot from simulation of shallow-water equations in which shock waves are present

130:), which hold even when the assumptions of shallow-water break down, such as across a 9025: 9010: 8858: 8778: 8667: 8586: 8561: 8496: 8426: 8333: 8228: 8105: 8026: 7986: 7959: 7869: 7819: 7574: 7287: 7219:"An analytical and numerical study of long wave run-up in U-shaped and V-shaped bays" 7203: 7095: 5758:{\displaystyle v{\frac {\partial u}{\partial y}}+w{\frac {\partial u}{\partial z}}=0} 3829: 3647: 2187: 1524: 631: 7154: 7103: 9005: 8965: 8913: 8853: 8804: 8682: 8677: 8652: 8636: 8611: 8328: 8218: 8158: 7944: 7854: 7829: 7472:

Computer applications in hydraulic engineering : connecting theory to practice

7305:. Mathematical Surveys and Monographs. American Mathematical Society. p. 174. 6698: 6629: 5986:

There are 2 body forces acting on the channel fluid, namely, gravity and friction:

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are functions of the channel geometry, described in the terms of the channel width

2215: 2203: 1463: 1414: 651: 52: 30: 4919: 8955: 8687: 8616: 8481: 8421: 8388: 8378: 8373: 8258: 8193: 8153: 8143: 8118: 8001: 7974: 7954: 7914: 7879: 6934: 6625: 6621: 3716: 3598: 3170:). Here σ is the height above the lowest point in the cross section at location 8773: 8621: 8596: 8491: 8471: 8398: 8383: 8368: 8358: 8323: 8243: 8063: 8058: 8021: 8016: 8011: 7909: 7271: 7235: 7218: 7186: 7169: 7138: 6859: 6764: 6710: 6690: 6686: 6665: 6633: 5508:

is the pressure, ρ is the density of water, ν is the kinematic viscosity, and

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is the total fluid column height (instantaneous fluid depth as a function of

8845: 8707: 8692: 8606: 8451: 8290: 8285: 8068: 7996: 7924: 7844: 7834: 7791: 6255:{\displaystyle \sin \theta =\tan \theta ={\frac {\text{opp}}{\text{adj}}}=S} 2279:. For example, for a rectangular cross section, with constant channel width 1516: 147: 7590:"A Shallow-Water Model that Prevents Nonlinear Steepening of Gravity Waves" 5523:

If it is assumed that friction is taken into account as a body force, then

7700: 6587:{\displaystyle (a)\quad \ \ (b)\quad \ \ \ (c)\qquad \ \ \ (d)\quad (e)\ } 8940: 8662: 8521: 8413: 8403: 8348: 7824: 7557: 7532: 6605: 5028: 2668:{\displaystyle S=-{\frac {\mathrm {d} z_{\mathrm {b} }}{\mathrm {d} x}},} 2569: 1865: 143: 7752: 6946: 6604:

constriction or an opening, respectively. Both these terms make up the

8809: 8799: 7969: 7939: 6661: 6653: 5090: 4930: 4915: 4450: 2936:, through some algebraic manipulations on the Saint-Venant equations, ( 2572:

equation, giving the balance between forces and momentum change rates.

2238: 1861: 1857: 1418: 639: 55:). The shallow-water equations in unidirectional form are also called 7662: 8516: 7929: 7531:

Augier, Pierre; Mohanan, Ashwin Vishnu; Lindborg, Erik (2019-09-17).

2359:) the bed level (i.e. elevation of the lowest point in the bed above 1447: 5135:. Statements consisting only of original research should be removed. 8878: 8697: 8476: 8431: 7320: 6676: 6648: 6159:

The expression for sin θ can be simplified using trigonometry as:

6151: 4934: 3657: 1879: 1484: 29: 8310: 7037:(1960), Theory of characteristics of inviscid gas dynamics. In: 5032: 4911: 3617:

gives the effects of geometry variations along the channel axis

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in atmospheric and oceanic modeling, as a simplification of the

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Derivation of the shallow-water equations from first principles

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The 1-D Saint-Venant momentum equation can be derived from the

5017:{\displaystyle g{\frac {\partial h}{\partial x}}+g(S_{f}-S)=0.} 4321:

In case there is no friction and the channel has a rectangular

2495:) the effective width of the channel cross section at location 109:

A one-dimensional diagram representing the shallow water model.

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The Water Waves Problem: Mathematical Analysis and Asymptotics

6846:; Yeh, H. (2005), "Tsunami propagation from a finite source", 6194:{\displaystyle \sin \theta ={\frac {\text{opp}}{\text{hyp}}}.} 5175:-component of the Navier–Stokes equations – when expressed in 5100: 3776:{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=u\pm c,} 6673:

Turbulence modelling using non-linear shallow-water equations

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waves in the atmosphere, rivers, lakes and oceans as well as

6205:(reasonable for almost all streams) it can be assumed that: 3363: 7615:

10.1175/1520-0469(1998)055<2884:ASWMTP>2.0.CO;2

1446:), where Ω is the angular rotation rate of the Earth (π/12 64: 6271:

represents a force per unit mass, the expression becomes:

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For a rectangular and prismatic channel of constant width

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The shallow-water equations are derived from equations of

6721:, avoids shock formation, but conserves only linearized 6156:

Figure 1: Diagram of block moving down an inclined plane.

2717:{\displaystyle S_{\mathrm {f} }={\frac {\tau }{\rho gR}}} 2367:). For non-moving channel walls the cross-sectional area 7639:"A two-dimensional toy model for geophysical turbulence" 6080:

can be calculated using basic physics and trigonometry:

1860:, SWMM5, ISIS, InfoWorks, Flood Modeller, SOBEK 1DFlow, 1519:, are small compared to the other terms. This is called 7511:, World Scientific, Singapore, pp. 473 & 516, 5124: 2476:{\displaystyle A(x,t)=\int _{0}^{h(x,t)}b(x,h')\,dh',} 1369:

is the topographical height from a reference D, where

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is the mean height of the horizontal pressure surface

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Lindborg, Erik; Mohanan, Ashwin Vishnu (2017-11-01).

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The equations are derived from depth-integrating the

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Cunge, J. A., F. M. Holly Jr. and A. Verwey (1980),

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Practical aspects of computational river hydraulics

6881:"Desktop review of 2D hydraulic modelling packages" 7364:. Computer models of watershed hydrology, 733–782. 6608:terms of the 1-dimensional Saint-Venant equation. 6586: 6494: 6362: 6308: 6254: 6193: 6123: 6043: 5976: 5838: 5791: 5757: 5679: 5535: 5466: 5078: 5016: 4868: 4429: 4300: 3996: 3944: 3818: 3775: 3532: 3120: 2897: 2752: 2716: 2667: 2475: 2122: 1980: 1802: 1242: 638:is acceleration due to gravity and ρ is the fluid 596: 8726:North West Shelf Operational Oceanographic System 6819:Clint Dawson and Christopher M. Mirabito (2008). 4335:is equal to the energy of the free-surface flow: 4325:cross section, the Saint-Venant equations have a 1507:It is often the case that the terms quadratic in 788: 763: 738: 713: 650:Expanding the derivatives in the above using the 7360:Yeh, G.; Cheng, J.; Lin, J.; Martin, W. (1995), 2150:is the space coordinate along the channel axis, 8716:Deep-ocean Assessment and Reporting of Tsunamis 6848:Computer Modeling in Engineering & Sciences 6689:in the atmosphere and oceans, i.e. geophysical 3646:, the conservation form is preferred since the 634:, averaged across the vertical column. Further 7174:Journal of Ocean Engineering and Marine Energy 7069: 7067: 5846:And when these assumptions are applied to the 5839:{\displaystyle \partial p=\rho g(\partial h).} 3642:). For instance in case of the description of 3189:) (of the lowest point in the cross section): 7768: 6930: 6928: 6887:(Science Report: SC080035): 5. Archived from 6620:Shallow-water equations can be used to model 4004:so the equations in characteristic form are: 3601:force in a certain cross section. And, for a 2275:and the surface elevation ζ at each position 1537:), we have (without lateral viscous forces): 8: 1819:one-dimensional (1-D) Saint-Venant equations 5850:-component of the Navier–Stokes equations: 4317:Hamiltonian structure for frictionless flow 1825:, and are commonly used to model transient 7775: 7761: 7753: 6668:, later on. The water depth is 100 meters. 3819:{\displaystyle c={\sqrt {\frac {gA}{B}}}.} 3509: 3377: 3263: 1840:The 1-D equations are used extensively in 7613: 7556: 7234: 7185: 6983: 6981: 6979: 6915:Comptes Rendus de l'Académie des Sciences 6616:Wave modelling by shallow-water equations 6506: 6468: 6435: 6409: 6383: 6381: 6348: 6329: 6323: 6282: 6276: 6236: 6210: 6178: 6164: 6091: 6085: 6029: 6010: 5997: 5991: 5951: 5906: 5896: 5880: 5860: 5855: 5804: 5772: 5729: 5703: 5698: 5657: 5639: 5632: 5620: 5602: 5595: 5583: 5565: 5558: 5548: 5528: 5455: 5434: 5416: 5409: 5397: 5379: 5372: 5360: 5342: 5335: 5314: 5294: 5268: 5242: 5216: 5190: 5188: 5151:Learn how and when to remove this message 5058: 5052: 4993: 4960: 4955: 4828: 4802: 4776: 4738: 4719: 4696: 4642: 4619: 4574: 4551: 4513: 4494: 4471: 4460: 4458: 4416: 4405: 4385: 4376: 4359: 4340: 4281: 4261: 4251: 4248: 4242: 4228: 4192: 4167: 4161: 4159: 4149: 4138: 4118: 4108: 4105: 4099: 4085: 4049: 4024: 4018: 4016: 4011: 4009: 3981: 3963: 3957: 3932: 3914: 3908: 3796: 3788: 3747: 3737: 3734: 3732: 3510: 3469: 3468: 3442: 3437: 3405: 3392: 3378: 3362: 3351: 3325: 3320: 3288: 3264: 3234: 3229: 3196: 3194: 3178:. So σ is the height above the bed level 3103: 3098: 3075: 3065: 3061: 3044: 3039: 3022: 3016: 2996: 2973: 2971: 2871: 2870: 2860: 2834: 2833: 2807: 2806: 2780: 2778: 2737: 2729: 2696: 2686: 2685: 2679: 2651: 2642: 2641: 2632: 2629: 2618: 2458: 2414: 2409: 2382: 2107: 2106: 2096: 2070: 2069: 2043: 2042: 2016: 2014: 1939: 1916: 1914: 1823:Adhémar Jean Claude Barré de Saint-Venant 1764: 1722: 1686: 1644: 1606: 1583: 1548: 1544: 1542: 1219: 1201: 1194: 1182: 1164: 1157: 1117: 1079: 1053: 1023: 1002: 984: 977: 965: 947: 940: 900: 862: 836: 806: 787: 786: 762: 761: 746: 737: 736: 712: 711: 696: 669: 665: 663: 549: 535: 515: 506: 476: 437: 389: 375: 355: 346: 316: 277: 232: 197: 161: 157: 155: 61:Adhémar Jean Claude Barré de Saint-Venant 49:hyperbolic partial differential equations 7719:Numerical Methods for Shallow-Water Flow 6756:Numerical Methods for Shallow-Water Flow 2537:, they can be related by using e.g. the 104: 7507:, Advanced Series on Ocean Engineering 6957: 6955: 6745: 5097:Derivation from Navier–Stokes equations 1523:, and is equivalent to saying that the 8047:one-dimensional Saint-Venant equations 3997:{\displaystyle r_{-}=u-2{\sqrt {gh}},} 3175: 2364: 1813:One-dimensional Saint-Venant equations 18:One-dimensional Saint-Venant equations 7408: 7406: 6124:{\displaystyle F_{g}=\sin(\theta )gM} 6058:is the body force due to gravity and 6044:{\displaystyle f_{x}=f_{x,g}+f_{x,f}} 5689:Assuming one-dimensional flow in the 3945:{\displaystyle r_{+}=u+2{\sqrt {gh}}} 2762:Consequently, the momentum equation ( 1515:, which represent the effect of bulk 7: 8994: 7505:Wave propagation over uneven bottoms 7415:Modeling and control of hydrosystems 6874: 6872: 6870: 6868: 2963: 2932:) can also be cast in the so-called 2770: 2237:. Further ρ is the (constant) fluid 2006: 1906: 7594:Journal of the Atmospheric Sciences 7223:Applied Mathematics and Computation 6697:, is that it allows solutions like 2314:. The instantaneous water depth is 150:, the shallow-water equations are: 8874:National Oceanographic Data Center 8301:World Ocean Circulation Experiment 8189:Global Ocean Data Analysis Project 6446: 6438: 6420: 6412: 6394: 6386: 6065:is the body force due to friction. 5962: 5954: 5933: 5920: 5871: 5863: 5824: 5806: 5740: 5732: 5714: 5706: 5650: 5636: 5613: 5599: 5576: 5562: 5427: 5413: 5390: 5376: 5353: 5339: 5305: 5297: 5279: 5271: 5253: 5245: 5227: 5219: 5201: 5193: 4971: 4963: 4839: 4831: 4813: 4805: 4787: 4779: 4749: 4741: 4725: 4721: 4707: 4699: 4662: 4645: 4630: 4622: 4594: 4577: 4562: 4554: 4524: 4516: 4500: 4496: 4482: 4474: 4417: 4262: 4252: 4168: 4162: 4119: 4109: 4025: 4019: 3748: 3738: 3727:on the characteristic curves are: 3542:Above – in the momentum equation ( 3511: 3500: 3472: 3379: 3265: 3002: 2998: 2984: 2976: 2872: 2845: 2837: 2818: 2810: 2791: 2783: 2687: 2652: 2643: 2633: 2533:is dependent on the flow velocity 2081: 2073: 2054: 2046: 2027: 2019: 1963: 1942: 1927: 1919: 1884:Cross section of the open channel. 1775: 1767: 1733: 1725: 1697: 1689: 1655: 1647: 1617: 1609: 1594: 1586: 1559: 1551: 1212: 1198: 1175: 1161: 1128: 1120: 1090: 1082: 1064: 1056: 1034: 1026: 995: 981: 958: 944: 911: 903: 873: 865: 847: 839: 817: 809: 752: 748: 702: 698: 680: 672: 575: 552: 482: 478: 460: 440: 415: 392: 322: 318: 300: 280: 255: 235: 220: 200: 181: 164: 25: 8721:Global Sea Level Observing System 4453:. Hamilton's equations then are: 2753:{\displaystyle R={\frac {A}{P}}.} 9004: 8993: 8984: 8983: 8179:Geochemical Ocean Sections Study 8095: 8084: 7039:Fluid Dynamics/Strömungsmechanik 5183:-direction – can be written as: 5105: 8909:Ocean thermal energy conversion 8632:Vine–Matthews–Morley hypothesis 7533:"Shallow water wave turbulence" 6571: 6552: 6533: 6517: 6363:{\displaystyle f_{x,f}=S_{f}g.} 6138:is the force of gravity in the 4247: 4148: 4104: 3845:determines whether the flow is 3650:is continuous across the jump. 3391: 2290:, the cross sectional area is: 124:conservation of linear momentum 9037:Partial differential equations 7721:, Kluwer Academic Publishers, 7695:, Cambridge University Press, 7693:Unsteady flow in open channels 7017:, Cambridge University Press, 6578: 6572: 6568: 6562: 6549: 6543: 6530: 6524: 6514: 6508: 6480: 6461: 6112: 6106: 5830: 5821: 5005: 4986: 4657: 4648: 4589: 4580: 3495: 3478: 3465: 3448: 3423: 3411: 3374: 3355: 3348: 3331: 3306: 3294: 3260: 3243: 3215: 3203: 2455: 2438: 2430: 2418: 2399: 2387: 2257:hyperbolic system of equations 1890:partial differential equations 1499:of Carrier and Yeh (2005) for 780: 768: 730: 718: 570: 555: 455: 443: 410: 395: 295: 283: 250: 238: 215: 203: 176: 167: 134:. In the case of a horizontal 1: 7588:Bühler, Oliver (1998-09-01). 6911:Saint-Venant, A.J.C. Barré de 6821:"The Shallow Water Equations" 6789:"The Shallow Water Equations" 6693:. An advantage of this, over 3636:), or the conservation form ( 72:Navier–Stokes equations 8169:El Niño–Southern Oscillation 8139:Craik–Leibovich vortex force 7895:Luke's variational principle 7749:, some analytical solutions) 7745:(instead of simplifying the 6994:, §§5.2 & 13.10, Wiley, 5693:-direction it follows that: 3711:) can be analysed using the 3699:The Saint-Venant equations ( 630:) is the fluid's horizontal 9032:Equations of fluid dynamics 7474:. Bentley Institute Press. 7252:Pure and Applied Geophysics 7119:Pure and Applied Geophysics 6879:S. Néelz; G Pender (2009). 6695:Quasi-geostrophic equations 6309:{\displaystyle f_{x,g}=gS.} 5543:can be assumed as zero so: 5131:the claims made and adding 4329:structure. The Hamiltonian 3707: 3701: 3638: 3632: 3626: 3544: 2944: 2938: 2928: 2764: 2564: 2554: 2373: 2267: 2261: 57:(de) Saint-Venant equations 9063: 8234:Ocean dynamical thermostat 8082: 7717:Vreugdenhil, C.B. (1994), 7537:Journal of Fluid Mechanics 7470:Methods., Haestad (2007). 7413:Vincent., Fromion (2009). 7096:10.1088/0951-7715/24/3/R01 7041:, Encyclopedia of Physics 6992:Linear and Nonlinear Waves 6753:Vreugdenhil, C.B. (1986). 5079:{\displaystyle S_{f}-S=0.} 2526:for rectangular channels. 2283:and channel bed elevation 2247:gravitational acceleration 2233:) of the cross section at 1469: 1457: 1423: 1406: 1302: 1292: 1274: 1256: 8979: 8769:Ocean acoustic tomography 8582:Mohorovičić discontinuity 8174:General circulation model 7810:Benjamin–Feir instability 7272:10.1007/s00024-017-1476-3 7236:10.1016/j.amc.2016.01.005 7187:10.1007/s40722-015-0040-4 7139:10.1007/s00024-014-1016-3 6860:10.3970/cmes.2005.010.113 6765:10.1007/978-94-015-8354-1 5799:or in differential form: 5792:{\displaystyle p=\rho gh} 5515:is the body force in the 3713:method of characteristics 3548:) in conservation form – 2166:) is the cross-sectional 1442:is equal to 2Ω sin( 8899:Ocean surface topography 8274:Thermohaline circulation 8264:Subsurface ocean current 8204:Hydrothermal circulation 8037:Wave–current interaction 7815:Boussinesq approximation 7691:; Labeur, R. J. (2017), 7503:Dingemans, M.W. (1997), 7451:www.physicsclassroom.com 6701:, while also conserving 6639:Boussinesq approximation 5031:and LISFLOOD-FP. In the 2922:Conservation of momentum 2499:when the fluid depth is 2170:of the flow at location 8936:Sea surface temperature 8919:Outline of oceanography 8114:Atmospheric circulation 8052:shallow water equations 8042:Waves and shallow water 7935:Significant wave height 7747:Navier–Stokes equations 6939:Open-channel hydraulics 6735:Waves and shallow water 5496:is the velocity in the 5488:is the velocity in the 5480:is the velocity in the 5165:Navier–Stokes equations 2926:The momentum equation ( 2539:Darcy–Weisbach equation 1892:which describe the 1-D 1281:is the velocity in the 1263:is the velocity in the 128:Navier–Stokes equations 41:shallow-water equations 8931:Sea surface microlayer 8296:Wind generated current 6682: 6669: 6588: 6496: 6364: 6310: 6256: 6195: 6157: 6125: 6045: 5978: 5840: 5793: 5759: 5681: 5537: 5468: 5080: 5018: 4870: 4443:the channel width and 4431: 4302: 3998: 3946: 3820: 3777: 3696: 3534: 3122: 2899: 2754: 2718: 2669: 2529:The wall shear stress 2477: 2124: 1982: 1885: 1804: 1504: 1244: 622:), and the 2D vector ( 598: 110: 36: 9042:Physical oceanography 8764:Deep scattering layer 8746:World Geodetic System 8254:Princeton Ocean Model 8134:Coriolis–Stokes force 7784:Physical oceanography 7701:10.1017/9781316576878 6965:, Pitman Publishing, 6921:: 147–154 and 237–240 6680: 6658:Boussinesq-type model 6652: 6589: 6497: 6365: 6311: 6257: 6196: 6155: 6126: 6046: 5979: 5841: 5794: 5760: 5682: 5538: 5469: 5177:Cartesian coordinates 5081: 5019: 4871: 4432: 4303: 3999: 3947: 3821: 3778: 3661: 3535: 3123: 2900: 2768:) can be written as: 2755: 2719: 2670: 2478: 2377:) can be written as: 2125: 1983: 1883: 1835:cross-sectional shape 1805: 1493: 1245: 646:Non-conservative form 599: 108: 93:of atmospheric flow. 33: 8784:Underwater acoustics 8344:Perigean spring tide 8209:Langmuir circulation 7920:Rossby-gravity waves 7558:10.1017/jfm.2019.375 7053:, Springer, Berlin, 6949:, §18-1 & §18-2. 6505: 6380: 6322: 6275: 6209: 6163: 6084: 5990: 5854: 5803: 5771: 5697: 5547: 5536:{\displaystyle \nu } 5527: 5187: 5051: 4954: 4457: 4339: 4008: 3956: 3907: 3787: 3731: 3193: 3176:cross-section figure 2970: 2777: 2728: 2678: 2617: 2381: 2365:cross-section figure 2013: 1913: 1541: 1434:associated with the 1432:Coriolis coefficient 662: 154: 120:conservation of mass 97:describe the state. 8946:Science On a Sphere 8552:Convergent boundary 8224:Modular Ocean Model 8184:Geostrophic current 7900:Mild-slope equation 7655:2017PhFl...29k1114L 7606:1998JAtS...55.2884B 7549:2019JFM...874.1169A 7301:Lannes, D. (2013). 7264:2017PApGe.174.3185A 7131:2015PApGe.172..885H 7088:2011Nonli..24R...1D 6891:on 8 September 2019 6723:potential vorticity 6707:potential vorticity 4449:the constant fluid 3447: 3330: 3239: 2948:). In terms of the 2560:continuity equation 2552:Further, equation ( 2434: 1894:incompressible flow 1521:geostrophic balance 1478:kinematic viscosity 91:primitive equations 8602:Seafloor spreading 8592:Outer trench swell 8557:Divergent boundary 8457:Continental margin 8442:Carbonate platform 8339:Lunitidal interval 7061:, pp. 225–282 6683: 6670: 6584: 6492: 6360: 6306: 6252: 6191: 6158: 6146:is the angle, and 6121: 6041: 5974: 5836: 5789: 5755: 5677: 5533: 5464: 5116:possibly contains 5076: 5014: 4925:2016-10-25 at the 4866: 4864: 4427: 4298: 4296: 3994: 3942: 3901:Riemann invariants 3816: 3773: 3697: 3530: 3528: 3433: 3316: 3225: 3118: 2895: 2750: 2714: 2665: 2613:) are defined as: 2583:), friction slope 2473: 2405: 2120: 1978: 1886: 1800: 1798: 1505: 1240: 1238: 594: 592: 138:, with negligible 111: 37: 9019: 9018: 9011:Oceans portal 8971:World Ocean Atlas 8961:Underwater glider 8904:Ocean temperature 8567:Hydrothermal vent 8532:Submarine volcano 8467:Continental shelf 8447:Coastal geography 8437:Bathymetric chart 8319:Amphidromic point 8007:Wave nonlinearity 7865:Infragravity wave 7710:978-1-107-15029-4 7663:10.1063/1.4985990 7643:Physics of Fluids 7600:(17): 2884–2891. 7518:978-981-02-0427-3 7447:"Inclined Planes" 7059:978-3-642-45946-7 7023:978-0-521-01045-0 6973:, §§2.1 & 2.2 6774:978-90-481-4472-3 6583: 6561: 6558: 6555: 6542: 6539: 6536: 6523: 6520: 6453: 6427: 6401: 6244: 6243: 6240: 6186: 6185: 6182: 5969: 5940: 5904: 5888: 5878: 5747: 5721: 5664: 5627: 5590: 5441: 5404: 5367: 5322: 5312: 5286: 5260: 5234: 5208: 5161: 5160: 5153: 5118:original research 4978: 4896:Derived modelling 4846: 4820: 4794: 4756: 4732: 4714: 4669: 4637: 4601: 4569: 4531: 4507: 4489: 4393: 4367: 4289: 4270: 4245: 4200: 4176: 4152: 4146: 4127: 4102: 4057: 4033: 3989: 3940: 3811: 3810: 3756: 3566:are evaluated at 3507: 3395: 3142: 3141: 3031: 3009: 2991: 2934:conservation form 2919: 2918: 2852: 2825: 2798: 2745: 2712: 2660: 2144: 2143: 2115: 2104: 2088: 2061: 2034: 2002: 2001: 1970: 1934: 1827:open-channel flow 1782: 1740: 1704: 1662: 1624: 1601: 1566: 1491: 1483: 1482: 1226: 1189: 1135: 1097: 1071: 1041: 1009: 972: 918: 880: 854: 824: 759: 709: 687: 582: 523: 489: 467: 422: 363: 329: 307: 262: 227: 188: 114:Conservative form 16:(Redirected from 9054: 9009: 9008: 8997: 8996: 8987: 8986: 8926:Pelagic sediment 8864:Marine pollution 8658:Deep ocean water 8527:Submarine canyon 8462:Continental rise 8354:Rule of twelfths 8269:Sverdrup balance 8199:Humboldt Current 8124:Boundary current 8099: 8088: 7905:Radiation stress 7875:Iribarren number 7850:Equatorial waves 7805:Ballantine scale 7800:Airy wave theory 7777: 7770: 7763: 7754: 7731: 7713: 7675: 7674: 7634: 7628: 7627: 7617: 7585: 7579: 7578: 7560: 7528: 7522: 7521: 7500: 7494: 7493: 7467: 7461: 7460: 7458: 7457: 7443: 7437: 7436: 7410: 7401: 7398: 7392: 7389: 7383: 7380: 7374: 7371: 7365: 7358: 7352: 7349: 7343: 7340: 7334: 7331: 7325: 7324: 7298: 7292: 7291: 7247: 7241: 7240: 7238: 7214: 7208: 7207: 7189: 7165: 7159: 7158: 7125:(3–4): 885–899. 7114: 7108: 7107: 7071: 7062: 7032: 7026: 7008: 7002: 6985: 6974: 6959: 6950: 6932: 6923: 6922: 6907: 6901: 6900: 6898: 6896: 6876: 6863: 6862: 6840: 6834: 6833: 6831: 6830: 6825: 6816: 6810: 6809: 6807: 6806: 6800: 6794:. Archived from 6793: 6785: 6779: 6778: 6750: 6593: 6591: 6590: 6585: 6581: 6559: 6556: 6553: 6540: 6537: 6534: 6521: 6518: 6501: 6499: 6498: 6493: 6473: 6472: 6454: 6452: 6444: 6436: 6428: 6426: 6418: 6410: 6402: 6400: 6392: 6384: 6369: 6367: 6366: 6361: 6353: 6352: 6340: 6339: 6315: 6313: 6312: 6307: 6293: 6292: 6261: 6259: 6258: 6253: 6245: 6241: 6238: 6237: 6200: 6198: 6197: 6192: 6187: 6183: 6180: 6179: 6130: 6128: 6127: 6122: 6096: 6095: 6050: 6048: 6047: 6042: 6040: 6039: 6021: 6020: 6002: 6001: 5983: 5981: 5980: 5975: 5970: 5968: 5960: 5952: 5941: 5939: 5931: 5930: 5926: 5907: 5905: 5897: 5889: 5881: 5879: 5877: 5869: 5861: 5845: 5843: 5842: 5837: 5798: 5796: 5795: 5790: 5764: 5762: 5761: 5756: 5748: 5746: 5738: 5730: 5722: 5720: 5712: 5704: 5686: 5684: 5683: 5678: 5670: 5666: 5665: 5663: 5662: 5661: 5648: 5644: 5643: 5633: 5628: 5626: 5625: 5624: 5611: 5607: 5606: 5596: 5591: 5589: 5588: 5587: 5574: 5570: 5569: 5559: 5542: 5540: 5539: 5534: 5473: 5471: 5470: 5465: 5460: 5459: 5447: 5443: 5442: 5440: 5439: 5438: 5425: 5421: 5420: 5410: 5405: 5403: 5402: 5401: 5388: 5384: 5383: 5373: 5368: 5366: 5365: 5364: 5351: 5347: 5346: 5336: 5323: 5315: 5313: 5311: 5303: 5295: 5287: 5285: 5277: 5269: 5261: 5259: 5251: 5243: 5235: 5233: 5225: 5217: 5209: 5207: 5199: 5191: 5156: 5149: 5145: 5142: 5136: 5133:inline citations 5109: 5108: 5101: 5085: 5083: 5082: 5077: 5063: 5062: 5023: 5021: 5020: 5015: 4998: 4997: 4979: 4977: 4969: 4961: 4933:, Wash 123d and 4891: 4875: 4873: 4872: 4867: 4865: 4852: 4848: 4847: 4845: 4837: 4829: 4821: 4819: 4811: 4803: 4795: 4793: 4785: 4777: 4761: 4757: 4755: 4747: 4739: 4733: 4731: 4720: 4715: 4713: 4705: 4697: 4688: 4675: 4671: 4670: 4668: 4660: 4643: 4638: 4636: 4628: 4620: 4607: 4603: 4602: 4600: 4592: 4575: 4570: 4568: 4560: 4552: 4536: 4532: 4530: 4522: 4514: 4508: 4506: 4495: 4490: 4488: 4480: 4472: 4463: 4448: 4442: 4436: 4434: 4433: 4428: 4420: 4415: 4411: 4410: 4409: 4394: 4386: 4381: 4380: 4368: 4360: 4334: 4307: 4305: 4304: 4299: 4297: 4290: 4282: 4271: 4269: 4265: 4259: 4255: 4249: 4246: 4243: 4240: 4238: 4234: 4233: 4232: 4206: 4202: 4201: 4193: 4177: 4175: 4171: 4165: 4160: 4157: 4153: 4150: 4147: 4139: 4128: 4126: 4122: 4116: 4112: 4106: 4103: 4100: 4097: 4095: 4091: 4090: 4089: 4063: 4059: 4058: 4050: 4034: 4032: 4028: 4022: 4017: 4014: 4003: 4001: 4000: 3995: 3990: 3982: 3968: 3967: 3951: 3949: 3948: 3943: 3941: 3933: 3919: 3918: 3898: 3897: 3896: 3883: 3866: 3855: 3844: 3825: 3823: 3822: 3817: 3812: 3806: 3798: 3797: 3782: 3780: 3779: 3774: 3757: 3755: 3751: 3745: 3741: 3735: 3686: 3616: 3596: 3584: 3539: 3537: 3536: 3531: 3529: 3522: 3514: 3508: 3506: 3498: 3488: 3470: 3464: 3446: 3441: 3410: 3409: 3396: 3393: 3390: 3382: 3367: 3366: 3347: 3329: 3324: 3293: 3292: 3276: 3268: 3253: 3238: 3233: 3136: 3127: 3125: 3124: 3119: 3108: 3107: 3091: 3087: 3080: 3079: 3054: 3050: 3049: 3048: 3032: 3027: 3026: 3017: 3010: 3008: 2997: 2992: 2990: 2982: 2974: 2964: 2960: 2913: 2904: 2902: 2901: 2896: 2888: 2884: 2877: 2876: 2875: 2853: 2851: 2843: 2835: 2826: 2824: 2816: 2808: 2799: 2797: 2789: 2781: 2771: 2759: 2757: 2756: 2751: 2746: 2738: 2723: 2721: 2720: 2715: 2713: 2711: 2697: 2692: 2691: 2690: 2674: 2672: 2671: 2666: 2661: 2659: 2655: 2649: 2648: 2647: 2646: 2636: 2630: 2600:hydraulic radius 2525: 2482: 2480: 2479: 2474: 2469: 2454: 2433: 2413: 2347: 2313: 2220:wetted perimeter 2206:elevation and τ( 2138: 2129: 2127: 2126: 2121: 2116: 2108: 2105: 2097: 2089: 2087: 2079: 2071: 2062: 2060: 2052: 2044: 2035: 2033: 2025: 2017: 2007: 1996: 1987: 1985: 1984: 1979: 1971: 1969: 1961: 1960: 1956: 1940: 1935: 1933: 1925: 1917: 1907: 1821:were derived by 1809: 1807: 1806: 1801: 1799: 1783: 1781: 1773: 1765: 1741: 1739: 1731: 1723: 1705: 1703: 1695: 1687: 1663: 1661: 1653: 1645: 1630: 1626: 1625: 1623: 1615: 1607: 1602: 1600: 1592: 1584: 1567: 1565: 1557: 1549: 1536: 1492: 1473: 1454:is the latitude 1403: 1359: 1255: 1254: 1249: 1247: 1246: 1241: 1239: 1232: 1228: 1227: 1225: 1224: 1223: 1210: 1206: 1205: 1195: 1190: 1188: 1187: 1186: 1173: 1169: 1168: 1158: 1136: 1134: 1126: 1118: 1098: 1096: 1088: 1080: 1072: 1070: 1062: 1054: 1042: 1040: 1032: 1024: 1015: 1011: 1010: 1008: 1007: 1006: 993: 989: 988: 978: 973: 971: 970: 969: 956: 952: 951: 941: 919: 917: 909: 901: 881: 879: 871: 863: 855: 853: 845: 837: 825: 823: 815: 807: 792: 791: 767: 766: 760: 758: 747: 742: 741: 717: 716: 710: 708: 697: 688: 686: 678: 670: 603: 601: 600: 595: 593: 583: 581: 573: 550: 545: 541: 540: 539: 524: 516: 511: 510: 490: 488: 477: 468: 466: 458: 438: 423: 421: 413: 390: 385: 381: 380: 379: 364: 356: 351: 350: 330: 328: 317: 308: 306: 298: 278: 263: 261: 253: 233: 228: 226: 218: 198: 189: 187: 179: 162: 21: 9062: 9061: 9057: 9056: 9055: 9053: 9052: 9051: 9022: 9021: 9020: 9015: 9003: 8975: 8814: 8788: 8750: 8731:Sea-level curve 8702: 8641: 8627:Transform fault 8577:Mid-ocean ridge 8543: 8536: 8502:Oceanic plateau 8408: 8394:Tidal resonance 8364:Theory of tides 8305: 8214:Longshore drift 8164:Ekman transport 8100: 8094: 8093: 8092: 8091: 8090: 8089: 8080: 8032:Wave turbulence 7965:Trochoidal wave 7890:Longshore drift 7786: 7781: 7739: 7734: 7729: 7716: 7711: 7687: 7683: 7681:Further reading 7678: 7636: 7635: 7631: 7587: 7586: 7582: 7530: 7529: 7525: 7519: 7502: 7501: 7497: 7482: 7469: 7468: 7464: 7455: 7453: 7445: 7444: 7440: 7425: 7412: 7411: 7404: 7399: 7395: 7390: 7386: 7381: 7377: 7372: 7368: 7359: 7355: 7350: 7346: 7341: 7337: 7332: 7328: 7313: 7300: 7299: 7295: 7249: 7248: 7244: 7216: 7215: 7211: 7167: 7166: 7162: 7116: 7115: 7111: 7073: 7072: 7065: 7033: 7029: 7015:Waves in fluids 7009: 7005: 6986: 6977: 6960: 6953: 6941:, McGraw-Hill, 6933: 6926: 6909: 6908: 6904: 6894: 6892: 6878: 6877: 6866: 6842: 6841: 6837: 6828: 6826: 6823: 6818: 6817: 6813: 6804: 6802: 6798: 6791: 6787: 6786: 6782: 6775: 6752: 6751: 6747: 6743: 6731: 6675: 6641:to incorporate 6618: 6503: 6502: 6464: 6445: 6437: 6419: 6411: 6393: 6385: 6378: 6377: 6344: 6325: 6320: 6319: 6278: 6273: 6272: 6270: 6262:and given that 6207: 6206: 6161: 6160: 6137: 6087: 6082: 6081: 6079: 6064: 6057: 6025: 6006: 5993: 5988: 5987: 5961: 5953: 5932: 5919: 5915: 5908: 5870: 5862: 5852: 5851: 5801: 5800: 5769: 5768: 5739: 5731: 5713: 5705: 5695: 5694: 5653: 5649: 5635: 5634: 5616: 5612: 5598: 5597: 5579: 5575: 5561: 5560: 5557: 5553: 5545: 5544: 5525: 5524: 5514: 5451: 5430: 5426: 5412: 5411: 5393: 5389: 5375: 5374: 5356: 5352: 5338: 5337: 5334: 5330: 5304: 5296: 5278: 5270: 5252: 5244: 5226: 5218: 5200: 5192: 5185: 5184: 5157: 5146: 5140: 5137: 5122: 5110: 5106: 5099: 5054: 5049: 5048: 5041: 4989: 4970: 4962: 4952: 4951: 4947: 4927:Wayback Machine 4903: 4898: 4877: 4863: 4862: 4838: 4830: 4812: 4804: 4786: 4778: 4775: 4771: 4748: 4740: 4734: 4724: 4706: 4698: 4686: 4685: 4661: 4644: 4629: 4621: 4618: 4614: 4593: 4576: 4561: 4553: 4547: 4543: 4523: 4515: 4509: 4499: 4481: 4473: 4455: 4454: 4444: 4438: 4401: 4372: 4358: 4354: 4337: 4336: 4330: 4319: 4295: 4294: 4260: 4250: 4239: 4224: 4217: 4213: 4182: 4178: 4166: 4155: 4154: 4117: 4107: 4096: 4081: 4074: 4070: 4039: 4035: 4023: 4006: 4005: 3959: 3954: 3953: 3910: 3905: 3904: 3892: 3890: 3885: 3875: 3861: 3850: 3832: 3799: 3785: 3784: 3746: 3736: 3729: 3728: 3684: 3675: 3663: 3656: 3654:Characteristics 3644:hydraulic jumps 3615: 3606: 3595: 3586: 3567: 3565: 3558: 3527: 3526: 3515: 3499: 3481: 3471: 3457: 3426: 3401: 3398: 3397: 3383: 3358: 3340: 3309: 3284: 3281: 3280: 3269: 3246: 3218: 3191: 3190: 3184: 3161: 3154: 3134: 3099: 3071: 3070: 3066: 3040: 3018: 3015: 3011: 3001: 2983: 2975: 2968: 2967: 2952: 2924: 2911: 2866: 2865: 2861: 2844: 2836: 2817: 2809: 2790: 2782: 2775: 2774: 2726: 2725: 2701: 2681: 2676: 2675: 2650: 2637: 2631: 2615: 2614: 2589: 2543:Manning formula 2504: 2462: 2447: 2379: 2378: 2354: 2341: 2315: 2305: 2291: 2289: 2136: 2080: 2072: 2053: 2045: 2026: 2018: 2011: 2010: 1994: 1962: 1949: 1945: 1941: 1926: 1918: 1911: 1910: 1878: 1842:computer models 1815: 1797: 1796: 1774: 1766: 1742: 1732: 1724: 1719: 1718: 1696: 1688: 1664: 1654: 1646: 1641: 1640: 1616: 1608: 1593: 1585: 1582: 1578: 1568: 1558: 1550: 1539: 1538: 1528: 1485: 1471: 1374: 1314: 1237: 1236: 1215: 1211: 1197: 1196: 1178: 1174: 1160: 1159: 1156: 1152: 1127: 1119: 1089: 1081: 1063: 1055: 1043: 1033: 1025: 1020: 1019: 998: 994: 980: 979: 961: 957: 943: 942: 939: 935: 910: 902: 872: 864: 846: 838: 826: 816: 808: 803: 802: 751: 701: 689: 679: 671: 660: 659: 648: 591: 590: 574: 551: 531: 502: 495: 491: 481: 469: 459: 439: 434: 433: 414: 391: 371: 342: 335: 331: 321: 309: 299: 279: 274: 273: 254: 234: 219: 199: 190: 180: 163: 152: 151: 140:Coriolis forces 116: 103: 87:Coriolis forces 65:related section 47:) are a set of 28: 23: 22: 15: 12: 11: 5: 9060: 9058: 9050: 9049: 9044: 9039: 9034: 9024: 9023: 9017: 9016: 9014: 9013: 9001: 8991: 8980: 8977: 8976: 8974: 8973: 8968: 8963: 8958: 8953: 8951:Stratification 8948: 8943: 8938: 8933: 8928: 8923: 8922: 8921: 8911: 8906: 8901: 8896: 8891: 8886: 8881: 8876: 8871: 8866: 8861: 8856: 8851: 8843: 8841:Color of water 8838: 8836:Benthic lander 8833: 8828: 8822: 8820: 8816: 8815: 8813: 8812: 8807: 8802: 8796: 8794: 8790: 8789: 8787: 8786: 8781: 8776: 8771: 8766: 8760: 8758: 8752: 8751: 8749: 8748: 8743: 8741:Sea level rise 8738: 8736:Sea level drop 8733: 8728: 8723: 8718: 8712: 8710: 8704: 8703: 8701: 8700: 8695: 8690: 8685: 8680: 8675: 8670: 8665: 8660: 8655: 8649: 8647: 8643: 8642: 8640: 8639: 8634: 8629: 8624: 8619: 8614: 8609: 8604: 8599: 8594: 8589: 8584: 8579: 8574: 8572:Marine geology 8569: 8564: 8559: 8554: 8548: 8546: 8538: 8537: 8535: 8534: 8529: 8524: 8519: 8514: 8512:Passive margin 8509: 8507:Oceanic trench 8504: 8499: 8494: 8489: 8484: 8479: 8474: 8469: 8464: 8459: 8454: 8449: 8444: 8439: 8434: 8429: 8424: 8418: 8416: 8410: 8409: 8407: 8406: 8401: 8396: 8391: 8386: 8381: 8376: 8371: 8366: 8361: 8356: 8351: 8346: 8341: 8336: 8331: 8326: 8321: 8315: 8313: 8307: 8306: 8304: 8303: 8298: 8293: 8288: 8283: 8282: 8281: 8271: 8266: 8261: 8256: 8251: 8246: 8241: 8239:Ocean dynamics 8236: 8231: 8226: 8221: 8216: 8211: 8206: 8201: 8196: 8191: 8186: 8181: 8176: 8171: 8166: 8161: 8156: 8151: 8146: 8141: 8136: 8131: 8129:Coriolis force 8126: 8121: 8116: 8110: 8108: 8102: 8101: 8083: 8081: 8079: 8078: 8077: 8076: 8066: 8061: 8056: 8055: 8054: 8049: 8039: 8034: 8029: 8024: 8019: 8014: 8009: 8004: 7999: 7994: 7989: 7984: 7979: 7978: 7977: 7967: 7962: 7957: 7952: 7950:Stokes problem 7947: 7942: 7937: 7932: 7927: 7922: 7917: 7912: 7907: 7902: 7897: 7892: 7887: 7885:Kinematic wave 7882: 7877: 7872: 7867: 7862: 7857: 7852: 7847: 7842: 7837: 7832: 7827: 7822: 7817: 7812: 7807: 7802: 7796: 7794: 7788: 7787: 7782: 7780: 7779: 7772: 7765: 7757: 7751: 7750: 7738: 7737:External links 7735: 7733: 7732: 7728:978-0792331643 7727: 7714: 7709: 7689:Battjes, J. A. 7684: 7682: 7679: 7677: 7676: 7649:(11): 111114. 7629: 7580: 7523: 7517: 7495: 7481:978-0971414167 7480: 7462: 7438: 7423: 7402: 7393: 7384: 7375: 7366: 7353: 7344: 7335: 7326: 7311: 7293: 7242: 7209: 7180:(2): 113–127. 7160: 7109: 7063: 7027: 7003: 6988:Whitham, G. B. 6975: 6951: 6924: 6902: 6864: 6854:(2): 113–122, 6844:Carrier, G. F. 6835: 6811: 6780: 6773: 6744: 6742: 6739: 6738: 6737: 6730: 6727: 6719:kinetic energy 6715:kinetic energy 6674: 6671: 6666:bore formation 6617: 6614: 6601: 6600: 6596: 6595: 6580: 6577: 6574: 6570: 6567: 6564: 6551: 6548: 6545: 6532: 6529: 6526: 6516: 6513: 6510: 6491: 6488: 6485: 6482: 6479: 6476: 6471: 6467: 6463: 6460: 6457: 6451: 6448: 6443: 6440: 6434: 6431: 6425: 6422: 6417: 6414: 6408: 6405: 6399: 6396: 6391: 6388: 6370: 6359: 6356: 6351: 6347: 6343: 6338: 6335: 6332: 6328: 6316: 6305: 6302: 6299: 6296: 6291: 6288: 6285: 6281: 6266: 6251: 6248: 6235: 6232: 6229: 6226: 6223: 6220: 6217: 6214: 6190: 6177: 6174: 6171: 6168: 6135: 6120: 6117: 6114: 6111: 6108: 6105: 6102: 6099: 6094: 6090: 6071: 6066: 6062: 6055: 6038: 6035: 6032: 6028: 6024: 6019: 6016: 6013: 6009: 6005: 6000: 5996: 5984: 5973: 5967: 5964: 5959: 5956: 5950: 5947: 5944: 5938: 5935: 5929: 5925: 5922: 5918: 5914: 5911: 5903: 5900: 5895: 5892: 5887: 5884: 5876: 5873: 5868: 5865: 5859: 5835: 5832: 5829: 5826: 5823: 5820: 5817: 5814: 5811: 5808: 5788: 5785: 5782: 5779: 5776: 5765: 5754: 5751: 5745: 5742: 5737: 5734: 5728: 5725: 5719: 5716: 5711: 5708: 5702: 5687: 5676: 5673: 5669: 5660: 5656: 5652: 5647: 5642: 5638: 5631: 5623: 5619: 5615: 5610: 5605: 5601: 5594: 5586: 5582: 5578: 5573: 5568: 5564: 5556: 5552: 5532: 5512: 5463: 5458: 5454: 5450: 5446: 5437: 5433: 5429: 5424: 5419: 5415: 5408: 5400: 5396: 5392: 5387: 5382: 5378: 5371: 5363: 5359: 5355: 5350: 5345: 5341: 5333: 5329: 5326: 5321: 5318: 5310: 5307: 5302: 5299: 5293: 5290: 5284: 5281: 5276: 5273: 5267: 5264: 5258: 5255: 5250: 5247: 5241: 5238: 5232: 5229: 5224: 5221: 5215: 5212: 5206: 5203: 5198: 5195: 5167:that describe 5159: 5158: 5113: 5111: 5104: 5098: 5095: 5075: 5072: 5069: 5066: 5061: 5057: 5045:kinematic wave 5040: 5039:Kinematic wave 5037: 5013: 5010: 5007: 5004: 5001: 4996: 4992: 4988: 4985: 4982: 4976: 4973: 4968: 4965: 4959: 4946: 4945:Diffusive wave 4943: 4902: 4899: 4897: 4894: 4861: 4858: 4855: 4851: 4844: 4841: 4836: 4833: 4827: 4824: 4818: 4815: 4810: 4807: 4801: 4798: 4792: 4789: 4784: 4781: 4774: 4770: 4767: 4764: 4760: 4754: 4751: 4746: 4743: 4737: 4730: 4727: 4723: 4718: 4712: 4709: 4704: 4701: 4695: 4692: 4689: 4687: 4684: 4681: 4678: 4674: 4667: 4664: 4659: 4656: 4653: 4650: 4647: 4641: 4635: 4632: 4627: 4624: 4617: 4613: 4610: 4606: 4599: 4596: 4591: 4588: 4585: 4582: 4579: 4573: 4567: 4564: 4559: 4556: 4550: 4546: 4542: 4539: 4535: 4529: 4526: 4521: 4518: 4512: 4505: 4502: 4498: 4493: 4487: 4484: 4479: 4476: 4470: 4467: 4464: 4462: 4437:with constant 4426: 4423: 4419: 4414: 4408: 4404: 4400: 4397: 4392: 4389: 4384: 4379: 4375: 4371: 4366: 4363: 4357: 4353: 4350: 4347: 4344: 4318: 4315: 4293: 4288: 4285: 4280: 4277: 4274: 4268: 4264: 4258: 4254: 4241: 4237: 4231: 4227: 4223: 4220: 4216: 4212: 4209: 4205: 4199: 4196: 4191: 4188: 4185: 4181: 4174: 4170: 4164: 4158: 4156: 4145: 4142: 4137: 4134: 4131: 4125: 4121: 4115: 4111: 4098: 4094: 4088: 4084: 4080: 4077: 4073: 4069: 4066: 4062: 4056: 4053: 4048: 4045: 4042: 4038: 4031: 4027: 4021: 4015: 4013: 3993: 3988: 3985: 3980: 3977: 3974: 3971: 3966: 3962: 3939: 3936: 3931: 3928: 3925: 3922: 3917: 3913: 3815: 3809: 3805: 3802: 3795: 3792: 3772: 3769: 3766: 3763: 3760: 3754: 3750: 3744: 3740: 3680: 3671: 3655: 3652: 3613: 3597:describes the 3593: 3563: 3556: 3525: 3521: 3518: 3513: 3505: 3502: 3497: 3494: 3491: 3487: 3484: 3480: 3477: 3474: 3467: 3463: 3460: 3456: 3453: 3450: 3445: 3440: 3436: 3432: 3429: 3427: 3425: 3422: 3419: 3416: 3413: 3408: 3404: 3400: 3399: 3389: 3386: 3381: 3376: 3373: 3370: 3365: 3361: 3357: 3354: 3350: 3346: 3343: 3339: 3336: 3333: 3328: 3323: 3319: 3315: 3312: 3310: 3308: 3305: 3302: 3299: 3296: 3291: 3287: 3283: 3282: 3279: 3275: 3272: 3267: 3262: 3259: 3256: 3252: 3249: 3245: 3242: 3237: 3232: 3228: 3224: 3221: 3219: 3217: 3214: 3211: 3208: 3205: 3202: 3199: 3198: 3182: 3159: 3152: 3140: 3139: 3130: 3128: 3117: 3114: 3111: 3106: 3102: 3097: 3094: 3090: 3086: 3083: 3078: 3074: 3069: 3064: 3060: 3057: 3053: 3047: 3043: 3038: 3035: 3030: 3025: 3021: 3014: 3007: 3004: 3000: 2995: 2989: 2986: 2981: 2978: 2923: 2920: 2917: 2916: 2907: 2905: 2894: 2891: 2887: 2883: 2880: 2874: 2869: 2864: 2859: 2856: 2850: 2847: 2842: 2839: 2832: 2829: 2823: 2820: 2815: 2812: 2805: 2802: 2796: 2793: 2788: 2785: 2749: 2744: 2741: 2736: 2733: 2710: 2707: 2704: 2700: 2695: 2689: 2684: 2664: 2658: 2654: 2645: 2640: 2635: 2628: 2625: 2622: 2587: 2575:The bed slope 2472: 2468: 2465: 2461: 2457: 2453: 2450: 2446: 2443: 2440: 2437: 2432: 2429: 2426: 2423: 2420: 2417: 2412: 2408: 2404: 2401: 2398: 2395: 2392: 2389: 2386: 2352: 2339: 2303: 2287: 2214:) is the wall 2154:denotes time, 2142: 2141: 2132: 2130: 2119: 2114: 2111: 2103: 2100: 2095: 2092: 2086: 2083: 2078: 2075: 2068: 2065: 2059: 2056: 2051: 2048: 2041: 2038: 2032: 2029: 2024: 2021: 2000: 1999: 1990: 1988: 1977: 1974: 1968: 1965: 1959: 1955: 1952: 1948: 1944: 1938: 1932: 1929: 1924: 1921: 1888:The system of 1877: 1874: 1831:surface runoff 1814: 1811: 1795: 1792: 1789: 1786: 1780: 1777: 1772: 1769: 1763: 1760: 1757: 1754: 1751: 1748: 1745: 1743: 1738: 1735: 1730: 1727: 1721: 1720: 1717: 1714: 1711: 1708: 1702: 1699: 1694: 1691: 1685: 1682: 1679: 1676: 1673: 1670: 1667: 1665: 1660: 1657: 1652: 1649: 1643: 1642: 1639: 1636: 1633: 1629: 1622: 1619: 1614: 1611: 1605: 1599: 1596: 1591: 1588: 1581: 1577: 1574: 1571: 1569: 1564: 1561: 1556: 1553: 1547: 1546: 1501:axisymmetrical 1497:exact solution 1481: 1480: 1474: 1468: 1467: 1460: 1456: 1455: 1436:Coriolis force 1428: 1422: 1421: 1411: 1405: 1404: 1367: 1361: 1360: 1307: 1301: 1300: 1297: 1291: 1290: 1285:direction, or 1279: 1273: 1272: 1267:direction, or 1261: 1235: 1231: 1222: 1218: 1214: 1209: 1204: 1200: 1193: 1185: 1181: 1177: 1172: 1167: 1163: 1155: 1151: 1148: 1145: 1142: 1139: 1133: 1130: 1125: 1122: 1116: 1113: 1110: 1107: 1104: 1101: 1095: 1092: 1087: 1084: 1078: 1075: 1069: 1066: 1061: 1058: 1052: 1049: 1046: 1044: 1039: 1036: 1031: 1028: 1022: 1021: 1018: 1014: 1005: 1001: 997: 992: 987: 983: 976: 968: 964: 960: 955: 950: 946: 938: 934: 931: 928: 925: 922: 916: 913: 908: 905: 899: 896: 893: 890: 887: 884: 878: 875: 870: 867: 861: 858: 852: 849: 844: 841: 835: 832: 829: 827: 822: 819: 814: 811: 805: 804: 801: 798: 795: 790: 785: 782: 779: 776: 773: 770: 765: 757: 754: 750: 745: 740: 735: 732: 729: 726: 723: 720: 715: 707: 704: 700: 695: 692: 690: 685: 682: 677: 674: 668: 667: 656:hydraulic jump 647: 644: 589: 586: 580: 577: 572: 569: 566: 563: 560: 557: 554: 548: 544: 538: 534: 530: 527: 522: 519: 514: 509: 505: 501: 498: 494: 487: 484: 480: 475: 472: 470: 465: 462: 457: 454: 451: 448: 445: 442: 436: 435: 432: 429: 426: 420: 417: 412: 409: 406: 403: 400: 397: 394: 388: 384: 378: 374: 370: 367: 362: 359: 354: 349: 345: 341: 338: 334: 327: 324: 320: 315: 312: 310: 305: 302: 297: 294: 291: 288: 285: 282: 276: 275: 272: 269: 266: 260: 257: 252: 249: 246: 243: 240: 237: 231: 225: 222: 217: 214: 211: 208: 205: 202: 196: 193: 191: 186: 183: 178: 175: 172: 169: 166: 160: 159: 148:viscous forces 132:hydraulic jump 115: 112: 102: 99: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 9059: 9048: 9045: 9043: 9040: 9038: 9035: 9033: 9030: 9029: 9027: 9012: 9007: 9002: 9000: 8992: 8990: 8982: 8981: 8978: 8972: 8969: 8967: 8964: 8962: 8959: 8957: 8954: 8952: 8949: 8947: 8944: 8942: 8939: 8937: 8934: 8932: 8929: 8927: 8924: 8920: 8917: 8916: 8915: 8912: 8910: 8907: 8905: 8902: 8900: 8897: 8895: 8892: 8890: 8887: 8885: 8882: 8880: 8877: 8875: 8872: 8870: 8867: 8865: 8862: 8860: 8859:Marine energy 8857: 8855: 8852: 8850: 8849: 8844: 8842: 8839: 8837: 8834: 8832: 8829: 8827: 8826:Acidification 8824: 8823: 8821: 8817: 8811: 8808: 8806: 8803: 8801: 8798: 8797: 8795: 8791: 8785: 8782: 8780: 8779:SOFAR channel 8777: 8775: 8772: 8770: 8767: 8765: 8762: 8761: 8759: 8757: 8753: 8747: 8744: 8742: 8739: 8737: 8734: 8732: 8729: 8727: 8724: 8722: 8719: 8717: 8714: 8713: 8711: 8709: 8705: 8699: 8696: 8694: 8691: 8689: 8686: 8684: 8681: 8679: 8676: 8674: 8671: 8669: 8666: 8664: 8661: 8659: 8656: 8654: 8651: 8650: 8648: 8644: 8638: 8635: 8633: 8630: 8628: 8625: 8623: 8620: 8618: 8615: 8613: 8610: 8608: 8605: 8603: 8600: 8598: 8595: 8593: 8590: 8588: 8587:Oceanic crust 8585: 8583: 8580: 8578: 8575: 8573: 8570: 8568: 8565: 8563: 8562:Fracture zone 8560: 8558: 8555: 8553: 8550: 8549: 8547: 8545: 8539: 8533: 8530: 8528: 8525: 8523: 8520: 8518: 8515: 8513: 8510: 8508: 8505: 8503: 8500: 8498: 8497:Oceanic basin 8495: 8493: 8490: 8488: 8485: 8483: 8480: 8478: 8475: 8473: 8470: 8468: 8465: 8463: 8460: 8458: 8455: 8453: 8450: 8448: 8445: 8443: 8440: 8438: 8435: 8433: 8430: 8428: 8427:Abyssal plain 8425: 8423: 8420: 8419: 8417: 8415: 8411: 8405: 8402: 8400: 8397: 8395: 8392: 8390: 8387: 8385: 8382: 8380: 8377: 8375: 8372: 8370: 8367: 8365: 8362: 8360: 8357: 8355: 8352: 8350: 8347: 8345: 8342: 8340: 8337: 8335: 8334:Internal tide 8332: 8330: 8327: 8325: 8322: 8320: 8317: 8316: 8314: 8312: 8308: 8302: 8299: 8297: 8294: 8292: 8289: 8287: 8284: 8280: 8277: 8276: 8275: 8272: 8270: 8267: 8265: 8262: 8260: 8257: 8255: 8252: 8250: 8247: 8245: 8242: 8240: 8237: 8235: 8232: 8230: 8229:Ocean current 8227: 8225: 8222: 8220: 8217: 8215: 8212: 8210: 8207: 8205: 8202: 8200: 8197: 8195: 8192: 8190: 8187: 8185: 8182: 8180: 8177: 8175: 8172: 8170: 8167: 8165: 8162: 8160: 8157: 8155: 8152: 8150: 8147: 8145: 8142: 8140: 8137: 8135: 8132: 8130: 8127: 8125: 8122: 8120: 8117: 8115: 8112: 8111: 8109: 8107: 8103: 8098: 8087: 8075: 8072: 8071: 8070: 8067: 8065: 8062: 8060: 8057: 8053: 8050: 8048: 8045: 8044: 8043: 8040: 8038: 8035: 8033: 8030: 8028: 8027:Wave shoaling 8025: 8023: 8020: 8018: 8015: 8013: 8010: 8008: 8005: 8003: 8000: 7998: 7995: 7993: 7990: 7988: 7987:Ursell number 7985: 7983: 7980: 7976: 7973: 7972: 7971: 7968: 7966: 7963: 7961: 7958: 7956: 7953: 7951: 7948: 7946: 7943: 7941: 7938: 7936: 7933: 7931: 7928: 7926: 7923: 7921: 7918: 7916: 7913: 7911: 7908: 7906: 7903: 7901: 7898: 7896: 7893: 7891: 7888: 7886: 7883: 7881: 7878: 7876: 7873: 7871: 7870:Internal wave 7868: 7866: 7863: 7861: 7858: 7856: 7853: 7851: 7848: 7846: 7843: 7841: 7838: 7836: 7833: 7831: 7828: 7826: 7823: 7821: 7820:Breaking wave 7818: 7816: 7813: 7811: 7808: 7806: 7803: 7801: 7798: 7797: 7795: 7793: 7789: 7785: 7778: 7773: 7771: 7766: 7764: 7759: 7758: 7755: 7748: 7744: 7741: 7740: 7736: 7730: 7724: 7720: 7715: 7712: 7706: 7702: 7698: 7694: 7690: 7686: 7685: 7680: 7672: 7668: 7664: 7660: 7656: 7652: 7648: 7644: 7640: 7633: 7630: 7625: 7621: 7616: 7611: 7607: 7603: 7599: 7595: 7591: 7584: 7581: 7576: 7572: 7568: 7564: 7559: 7554: 7550: 7546: 7543:: 1169–1196. 7542: 7538: 7534: 7527: 7524: 7520: 7514: 7510: 7506: 7499: 7496: 7491: 7487: 7483: 7477: 7473: 7466: 7463: 7452: 7448: 7442: 7439: 7434: 7430: 7426: 7424:9781848826243 7420: 7416: 7409: 7407: 7403: 7397: 7394: 7388: 7385: 7379: 7376: 7370: 7367: 7363: 7357: 7354: 7348: 7345: 7339: 7336: 7330: 7327: 7322: 7318: 7314: 7312:9780821894705 7308: 7304: 7297: 7294: 7289: 7285: 7281: 7277: 7273: 7269: 7265: 7261: 7257: 7253: 7246: 7243: 7237: 7232: 7228: 7224: 7220: 7213: 7210: 7205: 7201: 7197: 7193: 7188: 7183: 7179: 7175: 7171: 7164: 7161: 7156: 7152: 7148: 7144: 7140: 7136: 7132: 7128: 7124: 7120: 7113: 7110: 7105: 7101: 7097: 7093: 7089: 7085: 7082:(3): R1–R18. 7081: 7077: 7070: 7068: 7064: 7060: 7056: 7052: 7048: 7044: 7040: 7036: 7031: 7028: 7024: 7020: 7016: 7012: 7011:Lighthill, J. 7007: 7004: 7001: 7000:0-471-94090-9 6997: 6993: 6989: 6984: 6982: 6980: 6976: 6972: 6971:0 273 08442 9 6968: 6964: 6958: 6956: 6952: 6948: 6944: 6940: 6936: 6931: 6929: 6925: 6920: 6916: 6912: 6906: 6903: 6890: 6886: 6882: 6875: 6873: 6871: 6869: 6865: 6861: 6857: 6853: 6849: 6845: 6839: 6836: 6822: 6815: 6812: 6801:on 2012-03-16 6797: 6790: 6784: 6781: 6776: 6770: 6766: 6762: 6758: 6757: 6749: 6746: 6740: 6736: 6733: 6732: 6728: 6726: 6724: 6720: 6716: 6712: 6708: 6704: 6700: 6699:gravity waves 6696: 6692: 6688: 6679: 6672: 6667: 6663: 6659: 6655: 6651: 6647: 6644: 6640: 6635: 6631: 6630:gravity waves 6627: 6623: 6615: 6613: 6609: 6607: 6598: 6597: 6575: 6565: 6546: 6527: 6511: 6489: 6486: 6483: 6477: 6474: 6469: 6465: 6458: 6455: 6449: 6441: 6432: 6429: 6423: 6415: 6406: 6403: 6397: 6389: 6375: 6371: 6357: 6354: 6349: 6345: 6341: 6336: 6333: 6330: 6326: 6317: 6303: 6300: 6297: 6294: 6289: 6286: 6283: 6279: 6269: 6265: 6249: 6246: 6233: 6230: 6227: 6224: 6221: 6218: 6215: 6212: 6204: 6188: 6175: 6172: 6169: 6166: 6154: 6150:is the mass. 6149: 6145: 6141: 6134: 6118: 6115: 6109: 6103: 6100: 6097: 6092: 6088: 6078: 6074: 6070: 6067: 6061: 6054: 6036: 6033: 6030: 6026: 6022: 6017: 6014: 6011: 6007: 6003: 5998: 5994: 5985: 5971: 5965: 5957: 5948: 5945: 5942: 5936: 5927: 5923: 5916: 5912: 5909: 5901: 5898: 5893: 5890: 5885: 5882: 5874: 5866: 5857: 5849: 5833: 5827: 5818: 5815: 5812: 5809: 5786: 5783: 5780: 5777: 5774: 5766: 5752: 5749: 5743: 5735: 5726: 5723: 5717: 5709: 5700: 5692: 5688: 5674: 5671: 5667: 5658: 5654: 5645: 5640: 5629: 5621: 5617: 5608: 5603: 5592: 5584: 5580: 5571: 5566: 5554: 5550: 5530: 5522: 5521: 5520: 5518: 5511: 5507: 5503: 5499: 5495: 5491: 5487: 5484:-direction, 5483: 5479: 5474: 5461: 5456: 5452: 5448: 5444: 5435: 5431: 5422: 5417: 5406: 5398: 5394: 5385: 5380: 5369: 5361: 5357: 5348: 5343: 5331: 5327: 5324: 5319: 5316: 5308: 5300: 5291: 5288: 5282: 5274: 5265: 5262: 5256: 5248: 5239: 5236: 5230: 5222: 5213: 5210: 5204: 5196: 5182: 5178: 5174: 5170: 5166: 5155: 5152: 5144: 5134: 5130: 5126: 5120: 5119: 5114:This section 5112: 5103: 5102: 5096: 5094: 5092: 5086: 5073: 5070: 5067: 5064: 5059: 5055: 5046: 5038: 5036: 5034: 5030: 5024: 5011: 5008: 5002: 4999: 4994: 4990: 4983: 4980: 4974: 4966: 4957: 4944: 4942: 4938: 4936: 4932: 4928: 4924: 4921: 4920:InfoWorks_ICM 4917: 4913: 4909: 4900: 4895: 4893: 4889: 4885: 4881: 4859: 4856: 4853: 4849: 4842: 4834: 4825: 4822: 4816: 4808: 4799: 4796: 4790: 4782: 4772: 4768: 4765: 4762: 4758: 4752: 4744: 4735: 4728: 4716: 4710: 4702: 4693: 4690: 4682: 4679: 4676: 4672: 4665: 4654: 4651: 4639: 4633: 4625: 4615: 4611: 4608: 4604: 4597: 4586: 4583: 4571: 4565: 4557: 4548: 4544: 4540: 4537: 4533: 4527: 4519: 4510: 4503: 4491: 4485: 4477: 4468: 4465: 4452: 4447: 4441: 4424: 4421: 4412: 4406: 4402: 4398: 4395: 4390: 4387: 4382: 4377: 4373: 4369: 4364: 4361: 4355: 4351: 4348: 4345: 4342: 4333: 4328: 4324: 4316: 4314: 4311: 4308: 4291: 4286: 4283: 4278: 4275: 4272: 4266: 4256: 4235: 4229: 4225: 4221: 4218: 4214: 4210: 4207: 4203: 4197: 4194: 4189: 4186: 4183: 4179: 4172: 4143: 4140: 4135: 4132: 4129: 4123: 4113: 4092: 4086: 4082: 4078: 4075: 4071: 4067: 4064: 4060: 4054: 4051: 4046: 4043: 4040: 4036: 4029: 3991: 3986: 3983: 3978: 3975: 3972: 3969: 3964: 3960: 3937: 3934: 3929: 3926: 3923: 3920: 3915: 3911: 3902: 3895: 3888: 3882: 3878: 3873: 3868: 3864: 3859: 3858:supercritical 3853: 3848: 3843: 3839: 3835: 3831: 3830:Froude number 3826: 3813: 3807: 3803: 3800: 3793: 3790: 3770: 3767: 3764: 3761: 3758: 3752: 3742: 3726: 3722: 3718: 3714: 3710: 3709: 3704: 3703: 3694: 3690: 3683: 3679: 3674: 3670: 3666: 3660: 3653: 3651: 3649: 3648:momentum flux 3645: 3641: 3640: 3635: 3634: 3629: 3628: 3622: 3620: 3612: 3609: 3604: 3603:non-prismatic 3600: 3592: 3589: 3582: 3578: 3574: 3570: 3562: 3555: 3551: 3547: 3546: 3540: 3523: 3519: 3516: 3503: 3492: 3489: 3485: 3482: 3475: 3461: 3458: 3454: 3451: 3443: 3438: 3434: 3430: 3428: 3420: 3417: 3414: 3406: 3402: 3387: 3384: 3371: 3368: 3359: 3352: 3344: 3341: 3337: 3334: 3326: 3321: 3317: 3313: 3311: 3303: 3300: 3297: 3289: 3285: 3277: 3273: 3270: 3257: 3254: 3250: 3247: 3240: 3235: 3230: 3226: 3222: 3220: 3212: 3209: 3206: 3200: 3188: 3181: 3177: 3173: 3169: 3165: 3158: 3151: 3147: 3138: 3131: 3129: 3115: 3112: 3109: 3104: 3100: 3095: 3092: 3088: 3084: 3081: 3076: 3072: 3067: 3062: 3058: 3055: 3051: 3045: 3041: 3036: 3033: 3028: 3023: 3019: 3012: 3005: 2993: 2987: 2979: 2966: 2965: 2962: 2959: 2955: 2951: 2947: 2946: 2941: 2940: 2935: 2931: 2930: 2921: 2915: 2908: 2906: 2892: 2889: 2885: 2881: 2878: 2867: 2862: 2857: 2854: 2848: 2840: 2830: 2827: 2821: 2813: 2803: 2800: 2794: 2786: 2773: 2772: 2769: 2767: 2766: 2760: 2747: 2742: 2739: 2734: 2731: 2708: 2705: 2702: 2698: 2693: 2682: 2662: 2656: 2638: 2626: 2623: 2620: 2612: 2608: 2604: 2601: 2597: 2593: 2586: 2582: 2578: 2573: 2571: 2567: 2566: 2561: 2557: 2556: 2550: 2548: 2547:Chézy formula 2544: 2540: 2536: 2532: 2527: 2523: 2519: 2515: 2511: 2507: 2502: 2498: 2494: 2490: 2486: 2470: 2466: 2463: 2459: 2451: 2448: 2444: 2441: 2435: 2427: 2424: 2421: 2415: 2410: 2406: 2402: 2396: 2393: 2390: 2384: 2376: 2375: 2371:in equation ( 2370: 2366: 2362: 2358: 2351: 2345: 2338: 2334: 2330: 2326: 2322: 2318: 2312: 2309: 2302: 2298: 2294: 2286: 2282: 2278: 2274: 2270: 2269: 2264: 2263: 2258: 2254: 2250: 2248: 2244: 2240: 2236: 2232: 2228: 2224: 2221: 2217: 2213: 2209: 2205: 2201: 2197: 2193: 2189: 2188:flow velocity 2185: 2181: 2177: 2173: 2169: 2165: 2161: 2157: 2153: 2149: 2140: 2133: 2131: 2117: 2112: 2109: 2101: 2098: 2093: 2090: 2084: 2076: 2066: 2063: 2057: 2049: 2039: 2036: 2030: 2022: 2009: 2008: 2005: 1998: 1991: 1989: 1975: 1972: 1966: 1957: 1953: 1950: 1946: 1936: 1930: 1922: 1909: 1908: 1905: 1903: 1902:cross section 1900:of arbitrary 1899: 1895: 1891: 1882: 1875: 1873: 1871: 1870:flood routing 1867: 1863: 1859: 1855: 1851: 1847: 1843: 1838: 1836: 1832: 1828: 1824: 1820: 1812: 1810: 1793: 1790: 1787: 1784: 1778: 1770: 1761: 1758: 1755: 1752: 1749: 1746: 1744: 1736: 1728: 1715: 1712: 1709: 1706: 1700: 1692: 1683: 1680: 1677: 1674: 1671: 1668: 1666: 1658: 1650: 1637: 1634: 1631: 1627: 1620: 1612: 1603: 1597: 1589: 1579: 1575: 1572: 1570: 1562: 1554: 1535: 1531: 1526: 1525:Rossby number 1522: 1518: 1514: 1510: 1502: 1498: 1479: 1475: 1470: 1465: 1461: 1458: 1453: 1449: 1445: 1441: 1437: 1433: 1429: 1427: 1424: 1420: 1416: 1412: 1410: 1407: 1401: 1397: 1393: 1389: 1385: 1381: 1377: 1372: 1368: 1366: 1363: 1362: 1357: 1353: 1349: 1345: 1341: 1337: 1333: 1329: 1325: 1321: 1317: 1312: 1308: 1306: 1303: 1298: 1296: 1293: 1288: 1284: 1280: 1278: 1275: 1270: 1266: 1262: 1260: 1257: 1253: 1250: 1233: 1229: 1220: 1216: 1207: 1202: 1191: 1183: 1179: 1170: 1165: 1153: 1149: 1146: 1143: 1140: 1137: 1131: 1123: 1114: 1111: 1108: 1105: 1102: 1099: 1093: 1085: 1076: 1073: 1067: 1059: 1050: 1047: 1045: 1037: 1029: 1016: 1012: 1003: 999: 990: 985: 974: 966: 962: 953: 948: 936: 932: 929: 926: 923: 920: 914: 906: 897: 894: 891: 888: 885: 882: 876: 868: 859: 856: 850: 842: 833: 830: 828: 820: 812: 799: 796: 793: 783: 777: 774: 771: 755: 743: 733: 727: 724: 721: 705: 693: 691: 683: 675: 657: 653: 645: 643: 641: 637: 633: 632:flow velocity 629: 625: 621: 617: 613: 609: 604: 587: 584: 578: 567: 564: 561: 558: 546: 542: 536: 532: 528: 525: 520: 517: 512: 507: 503: 499: 496: 492: 485: 473: 471: 463: 452: 449: 446: 430: 427: 424: 418: 407: 404: 401: 398: 386: 382: 376: 372: 368: 365: 360: 357: 352: 347: 343: 339: 336: 332: 325: 313: 311: 303: 292: 289: 286: 270: 267: 264: 258: 247: 244: 241: 229: 223: 212: 209: 206: 194: 192: 184: 173: 170: 149: 145: 141: 137: 133: 129: 125: 121: 113: 107: 100: 98: 94: 92: 88: 83: 79: 77: 73: 68: 66: 62: 58: 54: 50: 46: 42: 32: 19: 8966:Water column 8914:Oceanography 8889:Observations 8884:Explorations 8854:Marginal sea 8847: 8805:OSTM/Jason-2 8637:Volcanic arc 8612:Slab suction 8329:Head of tide 8219:Loop Current 8159:Ekman spiral 8051: 7945:Stokes drift 7855:Gravity wave 7830:Cnoidal wave 7718: 7692: 7646: 7642: 7632: 7597: 7593: 7583: 7540: 7536: 7526: 7508: 7504: 7498: 7471: 7465: 7454:. Retrieved 7450: 7441: 7417:. Springer. 7414: 7396: 7387: 7378: 7369: 7361: 7356: 7347: 7338: 7329: 7302: 7296: 7255: 7251: 7245: 7226: 7222: 7212: 7177: 7173: 7163: 7122: 7118: 7112: 7079: 7076:Nonlinearity 7075: 7051:C. Truesdell 7042: 7038: 7035:Meyer, R. E. 7030: 7025:, §§2.8–2.14 7014: 7006: 6991: 6962: 6938: 6935:Chow, Ven Te 6918: 6914: 6905: 6893:. Retrieved 6889:the original 6884: 6851: 6847: 6838: 6827:. Retrieved 6814: 6803:. Retrieved 6796:the original 6783: 6755: 6748: 6684: 6619: 6610: 6602: 6376:-direction: 6373: 6267: 6263: 6202: 6147: 6143: 6142:-direction, 6139: 6132: 6076: 6072: 6068: 6059: 6052: 5847: 5690: 5519:-direction. 5516: 5509: 5505: 5501: 5500:-direction, 5497: 5493: 5492:-direction, 5489: 5485: 5481: 5477: 5475: 5180: 5172: 5169:fluid motion 5162: 5147: 5138: 5115: 5087: 5042: 5033:SIC (Irstea) 5025: 4948: 4939: 4912:SIC (Irstea) 4904: 4901:Dynamic wave 4887: 4883: 4879: 4445: 4439: 4331: 4320: 4312: 4309: 3893: 3886: 3880: 3876: 3874:, i.e. with 3871: 3869: 3862: 3851: 3841: 3837: 3833: 3827: 3724: 3720: 3706: 3700: 3698: 3692: 3688: 3681: 3677: 3672: 3668: 3664: 3637: 3631: 3625: 3623: 3618: 3610: 3607: 3590: 3587: 3580: 3576: 3572: 3568: 3560: 3553: 3549: 3543: 3541: 3186: 3179: 3171: 3167: 3163: 3156: 3149: 3145: 3143: 3132: 2957: 2953: 2943: 2937: 2927: 2925: 2909: 2763: 2761: 2610: 2606: 2602: 2595: 2591: 2584: 2580: 2576: 2574: 2563: 2553: 2551: 2534: 2530: 2528: 2521: 2517: 2513: 2509: 2505: 2500: 2496: 2492: 2488: 2484: 2372: 2368: 2356: 2349: 2343: 2336: 2332: 2328: 2324: 2320: 2316: 2310: 2307: 2300: 2296: 2292: 2284: 2280: 2276: 2272: 2266: 2260: 2251: 2242: 2234: 2230: 2226: 2222: 2216:shear stress 2211: 2207: 2204:free surface 2199: 2195: 2191: 2183: 2179: 2175: 2171: 2163: 2159: 2155: 2151: 2147: 2145: 2134: 2003: 1992: 1898:open channel 1887: 1854:SIC (Irstea) 1839: 1818: 1816: 1533: 1529: 1512: 1508: 1506: 1466:coefficient 1464:viscous drag 1451: 1450:/hour), and 1443: 1439: 1438:. On Earth, 1425: 1415:acceleration 1408: 1399: 1395: 1391: 1387: 1383: 1379: 1375: 1370: 1364: 1355: 1351: 1347: 1343: 1339: 1335: 1331: 1327: 1323: 1319: 1315: 1310: 1304: 1294: 1282: 1276: 1264: 1258: 1251: 652:product rule 649: 635: 627: 623: 619: 615: 611: 607: 605: 117: 95: 84: 80: 69: 56: 53:free surface 44: 40: 38: 9047:Water waves 8956:Thermocline 8673:Mesopelagic 8646:Ocean zones 8617:Slab window 8482:Hydrography 8422:Abyssal fan 8389:Tidal range 8379:Tidal power 8374:Tidal force 8259:Rip current 8194:Gulf Stream 8154:Ekman layer 8144:Downwelling 8119:Baroclinity 8106:Circulation 8002:Wave height 7992:Wave action 7975:megatsunami 7955:Stokes wave 7915:Rossby wave 7880:Kelvin wave 7860:Green's law 7258:(8): 3185. 7229:: 187–197. 6711:shock waves 4327:Hamiltonian 3847:subcritical 3599:hydrostatic 3585:. The term 76:hydrostatic 9026:Categories 8894:Reanalysis 8793:Satellites 8774:Sofar bomb 8622:Subduction 8597:Ridge push 8492:Ocean bank 8472:Contourite 8399:Tide gauge 8384:Tidal race 8369:Tidal bore 8359:Slack tide 8324:Earth tide 8244:Ocean gyre 8064:Wind setup 8059:Wind fetch 8022:Wave setup 8017:Wave radar 8012:Wave power 7910:Rogue wave 7840:Dispersion 7456:2017-05-16 7321:2012046540 6895:2 December 6829:2013-03-28 6805:2010-01-22 6691:turbulence 6687:turbulence 6643:dispersion 6634:wavelength 6201:For small 5141:April 2018 5125:improve it 3717:celerities 3715:. The two 3174:, see the 2363:, see the 2218:along the 1287:meridional 144:frictional 8756:Acoustics 8708:Sea level 8607:Slab pull 8544:tectonics 8452:Cold seep 8414:Landforms 8291:Whirlpool 8286:Upwelling 8069:Wind wave 7997:Wave base 7925:Sea state 7845:Edge wave 7835:Cross sea 7671:1070-6631 7624:0022-4928 7575:198976015 7567:1469-7645 7490:636350249 7433:401159458 7288:132114728 7280:0033-4553 7204:123725815 7196:2198-6444 7147:0033-4553 7047:S. Flügge 6475:− 6447:∂ 6439:∂ 6421:∂ 6413:∂ 6395:∂ 6387:∂ 6231:θ 6228:⁡ 6219:θ 6216:⁡ 6173:θ 6170:⁡ 6110:θ 6104:⁡ 5963:∂ 5955:∂ 5946:− 5934:∂ 5921:∂ 5910:ρ 5902:ρ 5894:− 5886:ρ 5872:∂ 5864:∂ 5858:− 5825:∂ 5816:ρ 5807:∂ 5781:ρ 5741:∂ 5733:∂ 5715:∂ 5707:∂ 5651:∂ 5637:∂ 5614:∂ 5600:∂ 5577:∂ 5563:∂ 5551:ν 5531:ν 5504:is time, 5428:∂ 5414:∂ 5391:∂ 5377:∂ 5354:∂ 5340:∂ 5328:ν 5320:ρ 5306:∂ 5298:∂ 5292:− 5280:∂ 5272:∂ 5254:∂ 5246:∂ 5228:∂ 5220:∂ 5202:∂ 5194:∂ 5129:verifying 5065:− 5000:− 4972:∂ 4964:∂ 4840:∂ 4835:ζ 4832:∂ 4814:∂ 4806:∂ 4788:∂ 4780:∂ 4766:ρ 4753:ζ 4750:∂ 4742:∂ 4726:∂ 4722:∂ 4708:∂ 4700:∂ 4691:ρ 4663:∂ 4646:∂ 4631:∂ 4623:∂ 4612:ρ 4595:∂ 4578:∂ 4563:∂ 4558:ζ 4555:∂ 4541:ρ 4525:∂ 4517:∂ 4501:∂ 4497:∂ 4483:∂ 4478:ζ 4475:∂ 4466:ρ 4403:ζ 4352:∫ 4349:ρ 4323:prismatic 4279:− 4222:− 4187:− 4079:− 3976:− 3965:− 3840:| / 3765:± 3691:and time 3687:in space 3605:channel, 3517:σ 3501:∂ 3483:σ 3473:∂ 3459:σ 3455:− 3452:σ 3444:σ 3435:∫ 3415:σ 3385:σ 3364:′ 3360:σ 3342:σ 3338:− 3335:σ 3327:σ 3318:∫ 3298:σ 3271:σ 3248:σ 3236:σ 3227:∫ 3207:σ 3093:− 3082:− 3003:∂ 2999:∂ 2985:∂ 2977:∂ 2950:discharge 2879:− 2846:∂ 2838:∂ 2819:∂ 2811:∂ 2792:∂ 2784:∂ 2703:ρ 2699:τ 2627:− 2568:) is the 2558:) is the 2407:∫ 2202:) is the 2186:) is the 2113:ρ 2110:τ 2094:− 2082:∂ 2077:ζ 2074:∂ 2055:∂ 2047:∂ 2028:∂ 2020:∂ 1964:∂ 1943:∂ 1928:∂ 1920:∂ 1876:Equations 1785:− 1776:∂ 1768:∂ 1759:− 1734:∂ 1726:∂ 1707:− 1698:∂ 1690:∂ 1681:− 1669:− 1656:∂ 1648:∂ 1618:∂ 1610:∂ 1595:∂ 1587:∂ 1560:∂ 1552:∂ 1517:advection 1289:velocity 1271:velocity 1213:∂ 1199:∂ 1176:∂ 1162:∂ 1150:ν 1138:− 1129:∂ 1121:∂ 1112:− 1091:∂ 1083:∂ 1065:∂ 1057:∂ 1035:∂ 1027:∂ 996:∂ 982:∂ 959:∂ 945:∂ 933:ν 921:− 912:∂ 904:∂ 895:− 883:− 874:∂ 866:∂ 848:∂ 840:∂ 818:∂ 810:∂ 753:∂ 749:∂ 703:∂ 699:∂ 681:∂ 673:∂ 576:∂ 562:η 559:ρ 553:∂ 533:η 526:ρ 500:η 497:ρ 483:∂ 479:∂ 461:∂ 450:η 447:ρ 441:∂ 416:∂ 402:η 399:ρ 393:∂ 373:η 366:ρ 340:η 337:ρ 323:∂ 319:∂ 301:∂ 290:η 287:ρ 281:∂ 256:∂ 245:η 242:ρ 236:∂ 221:∂ 210:η 207:ρ 201:∂ 182:∂ 174:η 171:ρ 165:∂ 101:Equations 63:(see the 8989:Category 8941:Seawater 8668:Littoral 8663:Deep sea 8522:Seamount 8404:Tideline 8349:Rip tide 8279:shutdown 8249:Overflow 7982:Undertow 7825:Clapotis 7155:55004099 7104:59438883 7013:(2005), 6937:(1959), 6729:See also 5043:For the 5029:MIKE SHE 4923:Archived 4908:Mascaret 3836:= | 3520:′ 3486:′ 3462:′ 3388:′ 3345:′ 3274:′ 3251:′ 2570:momentum 2467:′ 2452:′ 1866:MIKE SHE 1850:Mascaret 1844:such as 67:below). 59:, after 8999:Commons 8869:Mooring 8819:Related 8810:Jason-3 8800:Jason-1 8683:Pelagic 8678:Oceanic 8653:Benthic 7970:Tsunami 7940:Soliton 7651:Bibcode 7602:Bibcode 7545:Bibcode 7260:Bibcode 7127:Bibcode 7084:Bibcode 7045:, Eds. 6990:(1974) 6947:4010975 6662:soliton 6654:Tsunami 6606:inertia 5179:in the 5171:. The 5123:Please 5091:HEC-HMS 4931:MIKE 11 4916:HEC-RAS 4910:(EDF), 4451:density 3891:√ 2942:) and ( 2348:, with 2255:of the 2253:Closure 2245:is the 2239:density 1862:MIKE 11 1858:HEC-RAS 1852:(EDF), 1476:is the 1462:is the 1448:radians 1430:is the 1419:gravity 1417:due to 1413:is the 640:density 8688:Photic 8517:Seabed 7930:Seiche 7725: 7707: 7669: 7622: 7573: 7565: 7515: 7488: 7478: 7431: 7421: 7319: 7309: 7286: 7278: 7202: 7194: 7153: 7145: 7102: 7057: 7049:& 7021: 6998: 6969: 6945: 6771: 6703:energy 6626:Kelvin 6622:Rossby 6582: 6560: 6557: 6554: 6541: 6538: 6535: 6522: 6519: 6131:where 6051:where 5476:where 4876:since 3899:, the 3865:> 1 3854:< 1 3630:) or ( 3144:where 2598:) and 2327:) = ζ( 2146:where 1896:in an 1864:, and 1846:TUFLOW 1503:waves. 1252:where 608:η 35:walls. 8879:Ocean 8848:Alvin 8698:Swash 8542:Plate 8487:Knoll 8477:Guyot 8432:Atoll 8311:Tides 8074:model 7960:Swell 7792:Waves 7571:S2CID 7284:S2CID 7200:S2CID 7151:S2CID 7100:S2CID 6824:(PDF) 6799:(PDF) 6792:(PDF) 6741:Notes 6599:Terms 4935:SWMM5 4244:along 4101:along 3903:are: 3856:) or 3783:with 2503:– so 2483:with 2361:datum 2299:(ζ − 1269:zonal 606:Here 126:(the 8846:DSV 8831:Argo 8693:Surf 8149:Eddy 7723:ISBN 7705:ISBN 7667:ISSN 7620:ISSN 7563:ISSN 7513:ISBN 7486:OCLC 7476:ISBN 7429:OCLC 7419:ISBN 7317:LCCN 7307:ISBN 7276:ISSN 7192:ISSN 7143:ISSN 7055:ISBN 7019:ISBN 6996:ISBN 6967:ISBN 6943:OCLC 6897:2016 6769:ISBN 6705:and 6624:and 3952:and 3884:and 3828:The 3559:and 3155:and 2724:and 2516:) = 2335:) − 2306:) = 2241:and 2168:area 2004:and 1829:and 1817:The 1511:and 1386:) = 1342:) + 1330:) = 618:and 146:and 122:and 39:The 7697:doi 7659:doi 7610:doi 7553:doi 7541:874 7268:doi 7256:174 7231:doi 7227:279 7182:doi 7135:doi 7123:172 7092:doi 6856:doi 6761:doi 6242:adj 6239:opp 6225:tan 6213:sin 6184:hyp 6181:opp 6167:sin 6101:sin 6063:x,f 6056:x,g 5127:by 4151:and 3881:B h 3867:). 3705:)–( 3667:= ( 3394:and 3166:(σ, 2545:or 2265:)–( 136:bed 45:SWE 9028:: 7703:, 7665:. 7657:. 7647:29 7645:. 7641:. 7618:. 7608:. 7598:55 7596:. 7592:. 7569:. 7561:. 7551:. 7539:. 7535:. 7509:13 7484:. 7449:. 7427:. 7405:^ 7315:. 7282:. 7274:. 7266:. 7254:. 7225:. 7221:. 7198:. 7190:. 7176:. 7172:. 7149:. 7141:. 7133:. 7121:. 7098:. 7090:. 7080:24 7078:. 7066:^ 7043:IX 6978:^ 6954:^ 6927:^ 6919:73 6917:, 6883:. 6867:^ 6852:10 6850:, 6767:. 6725:. 5675:0. 5093:. 5074:0. 5012:0. 4937:. 4929:, 4918:, 4914:, 4892:. 4886:= 4882:/∂ 3894:gh 3889:= 3879:= 3863:Fr 3852:Fr 3834:Fr 3723:/d 3621:. 3571:= 3552:, 3148:, 2961:: 2958:Au 2956:= 2893:0. 2609:, 2594:, 2549:. 2541:, 2512:, 2295:= 2249:. 2190:, 2174:, 1856:, 1848:, 1837:. 1532:≪ 1390:+ 1382:, 1373:: 1354:, 1350:, 1338:, 1326:, 1322:, 1313:: 614:, 588:0. 142:, 7776:e 7769:t 7762:v 7699:: 7673:. 7661:: 7653:: 7626:. 7612:: 7604:: 7577:. 7555:: 7547:: 7492:. 7459:. 7435:. 7323:. 7290:. 7270:: 7262:: 7239:. 7233:: 7206:. 7184:: 7178:2 7157:. 7137:: 7129:: 7106:. 7094:: 7086:: 6899:. 6858:: 6832:. 6808:. 6777:. 6763:: 6579:) 6576:e 6573:( 6569:) 6566:d 6563:( 6550:) 6547:c 6544:( 6531:) 6528:b 6525:( 6515:) 6512:a 6509:( 6490:, 6487:0 6484:= 6481:) 6478:S 6470:f 6466:S 6462:( 6459:g 6456:+ 6450:x 6442:h 6433:g 6430:+ 6424:x 6416:u 6407:u 6404:+ 6398:t 6390:u 6374:x 6358:. 6355:g 6350:f 6346:S 6342:= 6337:f 6334:, 6331:x 6327:f 6304:. 6301:S 6298:g 6295:= 6290:g 6287:, 6284:x 6280:f 6268:x 6264:f 6250:S 6247:= 6234:= 6222:= 6203:θ 6189:. 6176:= 6148:M 6144:θ 6140:x 6136:g 6133:F 6119:M 6116:g 6113:) 6107:( 6098:= 6093:g 6089:F 6077:g 6075:, 6073:x 6069:f 6060:f 6053:f 6037:f 6034:, 6031:x 6027:f 6023:+ 6018:g 6015:, 6012:x 6008:f 6004:= 5999:x 5995:f 5972:. 5966:x 5958:h 5949:g 5943:= 5937:x 5928:) 5924:h 5917:( 5913:g 5899:1 5891:= 5883:1 5875:x 5867:p 5848:x 5834:. 5831:) 5828:h 5822:( 5819:g 5813:= 5810:p 5787:h 5784:g 5778:= 5775:p 5753:0 5750:= 5744:z 5736:u 5727:w 5724:+ 5718:y 5710:u 5701:v 5691:x 5672:= 5668:) 5659:2 5655:z 5646:u 5641:2 5630:+ 5622:2 5618:y 5609:u 5604:2 5593:+ 5585:2 5581:x 5572:u 5567:2 5555:( 5517:x 5513:x 5510:f 5506:p 5502:t 5498:z 5494:w 5490:y 5486:v 5482:x 5478:u 5462:, 5457:x 5453:f 5449:+ 5445:) 5436:2 5432:z 5423:u 5418:2 5407:+ 5399:2 5395:y 5386:u 5381:2 5370:+ 5362:2 5358:x 5349:u 5344:2 5332:( 5325:+ 5317:1 5309:x 5301:p 5289:= 5283:z 5275:u 5266:w 5263:+ 5257:y 5249:u 5240:v 5237:+ 5231:x 5223:u 5214:u 5211:+ 5205:t 5197:u 5181:x 5173:x 5154:) 5148:( 5143:) 5139:( 5121:. 5071:= 5068:S 5060:f 5056:S 5009:= 5006:) 5003:S 4995:f 4991:S 4987:( 4984:g 4981:+ 4975:x 4967:h 4958:g 4890:) 4888:B 4884:ζ 4880:A 4878:∂ 4860:, 4857:0 4854:= 4850:) 4843:x 4826:g 4823:+ 4817:x 4809:u 4800:u 4797:+ 4791:t 4783:u 4773:( 4769:B 4763:= 4759:) 4745:H 4736:( 4729:x 4717:+ 4711:t 4703:u 4694:B 4683:, 4680:0 4677:= 4673:) 4666:x 4658:) 4655:u 4652:A 4649:( 4640:+ 4634:t 4626:A 4616:( 4609:= 4605:) 4598:x 4590:) 4587:u 4584:A 4581:( 4572:+ 4566:t 4549:B 4545:( 4538:= 4534:) 4528:u 4520:H 4511:( 4504:x 4492:+ 4486:t 4469:B 4446:ρ 4440:B 4425:, 4422:x 4418:d 4413:) 4407:2 4399:B 4396:g 4391:2 4388:1 4383:+ 4378:2 4374:u 4370:A 4365:2 4362:1 4356:( 4346:= 4343:H 4332:H 4292:. 4287:h 4284:g 4276:u 4273:= 4267:t 4263:d 4257:x 4253:d 4236:) 4230:f 4226:S 4219:S 4215:( 4211:g 4208:= 4204:) 4198:h 4195:g 4190:2 4184:u 4180:( 4173:t 4169:d 4163:d 4144:h 4141:g 4136:+ 4133:u 4130:= 4124:t 4120:d 4114:x 4110:d 4093:) 4087:f 4083:S 4076:S 4072:( 4068:g 4065:= 4061:) 4055:h 4052:g 4047:2 4044:+ 4041:u 4037:( 4030:t 4026:d 4020:d 3992:, 3987:h 3984:g 3979:2 3973:u 3970:= 3961:r 3938:h 3935:g 3930:2 3927:+ 3924:u 3921:= 3916:+ 3912:r 3887:c 3877:A 3872:B 3860:( 3849:( 3842:c 3838:u 3814:. 3808:B 3804:A 3801:g 3794:= 3791:c 3771:, 3768:c 3762:u 3759:= 3753:t 3749:d 3743:x 3739:d 3725:t 3721:x 3719:d 3708:2 3702:1 3695:. 3693:t 3689:x 3685:) 3682:P 3678:t 3676:, 3673:P 3669:x 3665:P 3639:4 3633:3 3627:2 3619:x 3614:2 3611:I 3608:g 3594:1 3591:I 3588:g 3583:) 3581:t 3579:, 3577:x 3575:( 3573:h 3569:σ 3564:2 3561:I 3557:1 3554:I 3550:A 3545:4 3524:. 3512:d 3504:x 3496:) 3493:x 3490:, 3479:( 3476:B 3466:) 3449:( 3439:0 3431:= 3424:) 3421:x 3418:, 3412:( 3407:2 3403:I 3380:d 3375:) 3372:x 3369:, 3356:( 3353:B 3349:) 3332:( 3322:0 3314:= 3307:) 3304:x 3301:, 3295:( 3290:1 3286:I 3278:, 3266:d 3261:) 3258:x 3255:, 3244:( 3241:B 3231:0 3223:= 3216:) 3213:x 3210:, 3204:( 3201:A 3187:x 3185:( 3183:b 3180:z 3172:x 3168:x 3164:B 3160:2 3157:I 3153:1 3150:I 3146:A 3137:) 3135:4 3133:( 3116:, 3113:0 3110:= 3105:2 3101:I 3096:g 3089:) 3085:S 3077:f 3073:S 3068:( 3063:A 3059:g 3056:+ 3052:) 3046:1 3042:I 3037:g 3034:+ 3029:A 3024:2 3020:Q 3013:( 3006:x 2994:+ 2988:t 2980:Q 2954:Q 2945:3 2939:1 2929:3 2914:) 2912:3 2910:( 2890:= 2886:) 2882:S 2873:f 2868:S 2863:( 2858:g 2855:+ 2849:x 2841:h 2831:g 2828:+ 2822:x 2814:u 2804:u 2801:+ 2795:t 2787:u 2765:2 2748:. 2743:P 2740:A 2735:= 2732:R 2709:R 2706:g 2694:= 2688:f 2683:S 2663:, 2657:x 2653:d 2644:b 2639:z 2634:d 2624:= 2621:S 2611:t 2607:x 2605:( 2603:R 2596:t 2592:x 2590:( 2588:f 2585:S 2581:x 2579:( 2577:S 2565:2 2555:1 2535:u 2531:τ 2524:) 2522:x 2520:( 2518:B 2514:h 2510:x 2508:( 2506:b 2501:h 2497:x 2493:h 2491:, 2489:x 2487:( 2485:b 2471:, 2464:h 2460:d 2456:) 2449:h 2445:, 2442:x 2439:( 2436:b 2431:) 2428:t 2425:, 2422:x 2419:( 2416:h 2411:0 2403:= 2400:) 2397:t 2394:, 2391:x 2388:( 2385:A 2374:1 2369:A 2357:x 2355:( 2353:b 2350:z 2346:) 2344:x 2342:( 2340:b 2337:z 2333:t 2331:, 2329:x 2325:t 2323:, 2321:x 2319:( 2317:h 2311:h 2308:B 2304:b 2301:z 2297:B 2293:A 2288:b 2285:z 2281:B 2277:x 2273:A 2268:2 2262:1 2259:( 2243:g 2235:x 2231:t 2229:, 2227:x 2225:( 2223:P 2212:t 2210:, 2208:x 2200:t 2198:, 2196:x 2194:( 2192:ζ 2184:t 2182:, 2180:x 2178:( 2176:u 2172:x 2164:t 2162:, 2160:x 2158:( 2156:A 2152:t 2148:x 2139:) 2137:2 2135:( 2118:, 2102:A 2099:P 2091:= 2085:x 2067:g 2064:+ 2058:x 2050:u 2040:u 2037:+ 2031:t 2023:u 1997:) 1995:1 1993:( 1976:0 1973:= 1967:x 1958:) 1954:u 1951:A 1947:( 1937:+ 1931:t 1923:A 1794:. 1791:v 1788:k 1779:y 1771:h 1762:g 1756:= 1753:u 1750:f 1747:+ 1737:t 1729:v 1716:, 1713:u 1710:k 1701:x 1693:h 1684:g 1678:= 1675:v 1672:f 1659:t 1651:u 1638:, 1635:0 1632:= 1628:) 1621:y 1613:v 1604:+ 1598:x 1590:u 1580:( 1576:H 1573:+ 1563:t 1555:h 1534:H 1530:h 1513:v 1509:u 1472:ν 1459:k 1452:φ 1444:φ 1440:f 1426:f 1409:g 1402:) 1400:y 1398:, 1396:x 1394:( 1392:b 1388:D 1384:y 1380:x 1378:( 1376:H 1371:b 1365:b 1358:) 1356:t 1352:y 1348:x 1346:( 1344:h 1340:y 1336:x 1334:( 1332:H 1328:t 1324:y 1320:x 1318:( 1316:η 1311:h 1305:h 1295:H 1283:y 1277:v 1265:x 1259:u 1234:, 1230:) 1221:2 1217:y 1208:v 1203:2 1192:+ 1184:2 1180:x 1171:v 1166:2 1154:( 1147:+ 1144:v 1141:k 1132:y 1124:h 1115:g 1109:= 1106:u 1103:f 1100:+ 1094:y 1086:v 1077:v 1074:+ 1068:x 1060:v 1051:u 1048:+ 1038:t 1030:v 1017:, 1013:) 1004:2 1000:y 991:u 986:2 975:+ 967:2 963:x 954:u 949:2 937:( 930:+ 927:u 924:k 915:x 907:h 898:g 892:= 889:v 886:f 877:y 869:u 860:v 857:+ 851:x 843:u 834:u 831:+ 821:t 813:u 800:, 797:0 794:= 789:) 784:v 781:) 778:h 775:+ 772:H 769:( 764:( 756:y 744:+ 739:) 734:u 731:) 728:h 725:+ 722:H 719:( 714:( 706:x 694:+ 684:t 676:h 636:g 628:v 626:, 624:u 620:t 616:y 612:x 585:= 579:x 571:) 568:v 565:u 556:( 547:+ 543:) 537:2 529:g 521:2 518:1 513:+ 508:2 504:v 493:( 486:y 474:+ 464:t 456:) 453:v 444:( 431:, 428:0 425:= 419:y 411:) 408:v 405:u 396:( 387:+ 383:) 377:2 369:g 361:2 358:1 353:+ 348:2 344:u 333:( 326:x 314:+ 304:t 296:) 293:u 284:( 271:, 268:0 265:= 259:y 251:) 248:v 239:( 230:+ 224:x 216:) 213:u 204:( 195:+ 185:t 177:) 168:( 43:( 20:)

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Index