Shallow water equations - Knowledge (XXG)

6142: 1237: 650: 4863: 591: 4445: 1232:{\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}}} 142: 4858:{\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}}} 586:{\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}}} 4295: 3527: 6667: 3996: 5461: 1870: 3648: 1797: 8086: 3181: 5175: 8974: 4290:{\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}}} 1529: 8995: 95: 5096: 3522:{\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}}} 8075: 8984: 5456:{\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},} 63:, 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 1792:{\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}}} 67:, 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. 5971: 2117: 1475: 3115: 85:

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.

6489: 5842: 20: 1822:. 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 5015:

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.

6141: 643:, 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 4302:

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.

2711: 6702:. 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 74:

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".

3813: 1901: 6698:. 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 3991: 1485: 6118: 6038: 3939: 2747: 1861:

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

6357: 5966:{\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}}.} 6639: 6583:

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.

5685: 6303: 8714: 5073: 5786: 2112:{\displaystyle {\frac {\partial u}{\partial t}}+u\,{\frac {\partial u}{\partial x}}+g\,{\frac {\partial \zeta }{\partial x}}=-{\frac {P}{A}}\,{\frac {\tau }{\rho }},} 8704: 7763: 6899: 1811: 49: 5530: 6197: 6493: 6307:

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:

3110:{\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,} 2605: 5669:{\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.} 4299:

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

2887:{\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.} 7389:

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, (

8035: 7803: 6646: 6627: 2666: 6902:(1871), "Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et a l'introduction de marées dans leurs lits", 7697: 7505: 7371:

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.

7047: 7011: 6777: 6761: 2241: 6959: 2369: 40:(or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a 9020: 8267: 8157: 7756: 8862: 8289: 8177: 7715: 7468: 8709: 7980: 7411: 7299: 6988: 5791: 5139: 8127: 5106: 6484:{\displaystyle {\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}+g{\frac {\partial h}{\partial x}}+g(S_{f}-S)=0,} 8167: 5024:

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

647:. Also included are the appropriate terms for Coriolis, frictional and viscous forces, to obtain (for constant fluid density): 8570: 6638: 9030: 8977: 7749: 3944: 2260:) is obtained from the geometry of cross sections – by providing a functional relationship between the cross-sectional area 6072: 5978: 3895: 7023: 1878: 8025: 4911: 7735: 6653:

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.

7159:"Run-up of nonlinear long waves in U-shaped bays of finite length: analytical theory and numerical computations" 5039: 8887: 8262: 8252: 8192: 7828: 7798: 6631: 5759: 1890: 1823: 112: 6666: 8924: 8907: 8744: 8237: 8102: 8030: 7923: 6723: 6706:. Another option is to modify the non-linear terms in all equations, which gives a quadratic expression for 6874:

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

7435: 8752: 8734: 8242: 8137: 7772: 4315: 2938: 4419:{\displaystyle H=\rho \int \left({\frac {1}{2}}Au^{2}+{\frac {1}{2}}gB\zeta ^{2}\right)\mathrm {d} x,} 9035: 8939: 8772: 8475: 8332: 8197: 7908: 7639: 7590: 7533: 7248: 7115: 7072: 1420: 1275: 1257: 108: 64: 631:. The first equation is derived from mass conservation, the second two from momentum conservation. 8934: 8819: 8814: 8540: 8212: 8172: 7888: 7035: 6711: 6695: 6649:(blue line; with frequency dispersion). Observe that the Boussinesq-type model (blue line) forms a 4896: 2548: 1882: 1858: 1838: 1509: 1466: 79: 6784: 8877: 8590: 8580: 8545: 8445: 8430: 8327: 7559: 7272: 7206:

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

7188: 7139: 7088: 7039: 3846: 3835: 1970:{\displaystyle {\frac {\partial A}{\partial t}}+{\frac {\partial \left(Au\right)}{\partial x}}=0} 7362:

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

3647: 2551:, expressing conservation of water volume for this incompressible homogeneous fluid. Equation ( 8959: 8949: 8892: 8872: 8555: 8520: 8455: 8435: 8425: 8307: 7995: 7853: 7711: 7693: 7655: 7608: 7551: 7501: 7474: 7464: 7417: 7407: 7305: 7295: 7264: 7180: 7157:

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

5033: 3632: 3591: 2349: 1893:– as derived and posed by Saint-Venant in his 1871 paper (equations 19 & 20) – is: 1830: 1819: 1489: 1424: 1298:

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

119:), which hold even when the assumptions of shallow-water break down, such as across a 9014: 8999: 8847: 8767: 8656: 8575: 8550: 8485: 8415: 8322: 8217: 8094: 8015: 7975: 7948: 7858: 7808: 7563: 7276: 7208:"An analytical and numerical study of long wave run-up in U-shaped and V-shaped bays" 7192: 7084: 5747:{\displaystyle v{\frac {\partial u}{\partial y}}+w{\frac {\partial u}{\partial z}}=0} 3818: 3636: 2176: 1513: 620: 7143: 7092: 8994: 8954: 8902: 8842: 8793: 8671: 8666: 8641: 8625: 8600: 8317: 8207: 8147: 7933: 7843: 7818: 7461:

Computer applications in hydraulic engineering : connecting theory to practice

7294:. Mathematical Surveys and Monographs. American Mathematical Society. p. 174. 6687: 6618: 5975:

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

2204: 2192: 1452: 1403: 640: 41: 19: 4908: 8944: 8676: 8605: 8470: 8410: 8377: 8367: 8362: 8247: 8182: 8142: 8132: 8107: 7990: 7963: 7943: 7903: 7868: 6923: 6614: 6610: 3705: 3587: 3159:). Here σ is the height above the lowest point in the cross section at location 8762: 8610: 8585: 8480: 8460: 8387: 8372: 8357: 8347: 8312: 8232: 8052: 8047: 8010: 8005: 8000: 7898: 7260: 7224: 7207: 7175: 7158: 7127: 6848: 6753: 6699: 6679: 6675: 6654: 6622: 5497:

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

8834: 8696: 8681: 8595: 8440: 8279: 8274: 8057: 7985: 7913: 7833: 7823: 7780: 6244:{\displaystyle \sin \theta =\tan \theta ={\frac {\text{opp}}{\text{adj}}}=S} 2268:. For example, for a rectangular cross section, with constant channel width 1505: 136: 7579:"A Shallow-Water Model that Prevents Nonlinear Steepening of Gravity Waves" 5512:

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

7689: 6576:{\displaystyle (a)\quad \ \ (b)\quad \ \ \ (c)\qquad \ \ \ (d)\quad (e)\ } 8929: 8651: 8510: 8402: 8392: 8337: 7813: 7546: 7521: 6594: 5017: 2657:{\displaystyle S=-{\frac {\mathrm {d} z_{\mathrm {b} }}{\mathrm {d} x}},} 2558: 1854: 132: 7741: 6935: 6593:

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

8798: 8788: 7958: 7928: 6650: 6642: 5079: 4919: 4904: 4439: 2925:, through some algebraic manipulations on the Saint-Venant equations, ( 2561:

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

2227: 1850: 1846: 1407: 628: 44:). The shallow-water equations in unidirectional form are also called 7651: 8505: 7918: 7520:

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

2348:) the bed level (i.e. elevation of the lowest point in the bed above 1436: 5124:. Statements consisting only of original research should be removed. 8867: 8686: 8465: 8420: 7309: 6665: 6637: 6148:

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

6140: 4923: 3646: 1868: 1473: 18: 8299: 7026:(1960), Theory of characteristics of inviscid gas dynamics. In: 5021: 4900: 3606:

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

5006:{\displaystyle g{\frac {\partial h}{\partial x}}+g(S_{f}-S)=0.} 4310:

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

2484:) the effective width of the channel cross section at location 98:

A one-dimensional diagram representing the shallow water model.

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The Water Waves Problem: Mathematical Analysis and Asymptotics

6835:; Yeh, H. (2005), "Tsunami propagation from a finite source", 6183:{\displaystyle \sin \theta ={\frac {\text{opp}}{\text{hyp}}}.} 5164:-component of the Navier–Stokes equations – when expressed in 5089: 3765:{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=u\pm c,} 6662:

Turbulence modelling using non-linear shallow-water equations

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

6194:(reasonable for almost all streams) it can be assumed that: 3352: 7604:

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

1435:), where Ω is the angular rotation rate of the Earth (π/12 53: 6260:

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

6710:, avoids shock formation, but conserves only linearized 6145:

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

2706:{\displaystyle S_{\mathrm {f} }={\frac {\tau }{\rho gR}}} 2356:). For non-moving channel walls the cross-sectional area 7628:"A two-dimensional toy model for geophysical turbulence" 6069:

can be calculated using basic physics and trigonometry:

1849:, SWMM5, ISIS, InfoWorks, Flood Modeller, SOBEK 1DFlow, 1508:, are small compared to the other terms. This is called 7500:, World Scientific, Singapore, pp. 473 & 516, 5113: 2465:{\displaystyle A(x,t)=\int _{0}^{h(x,t)}b(x,h')\,dh',} 1358:

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

6870:"Desktop review of 2D hydraulic modelling packages" 7353:. Computer models of watershed hydrology, 733–782. 6597:terms of the 1-dimensional Saint-Venant equation. 6575: 6483: 6351: 6297: 6243: 6182: 6112: 6032: 5965: 5827: 5780: 5746: 5668: 5524: 5455: 5067: 5005: 4857: 4418: 4289: 3985: 3933: 3807: 3764: 3521: 3109: 2886: 2741: 2705: 2656: 2464: 2111: 1969: 1791: 1231: 627:is acceleration due to gravity and ρ is the fluid 585: 8715:North West Shelf Operational Oceanographic System 6808:Clint Dawson and Christopher M. Mirabito (2008). 4324:is equal to the energy of the free-surface flow: 4314:cross section, the Saint-Venant equations have a 1496:It is often the case that the terms quadratic in 777: 752: 727: 702: 639:Expanding the derivatives in the above using the 7349:Yeh, G.; Cheng, J.; Lin, J.; Martin, W. (1995), 2139:is the space coordinate along the channel axis, 8705:Deep-ocean Assessment and Reporting of Tsunamis 6837:Computer Modeling in Engineering & Sciences 6678:in the atmosphere and oceans, i.e. geophysical 3635:, the conservation form is preferred since the 623:, averaged across the vertical column. Further 7163:Journal of Ocean Engineering and Marine Energy 7058: 7056: 5835:And when these assumptions are applied to the 5828:{\displaystyle \partial p=\rho g(\partial h).} 3631:). For instance in case of the description of 3178:) (of the lowest point in the cross section): 7757: 6919: 6917: 6876:(Science Report: SC080035): 5. Archived from 6609:Shallow-water equations can be used to model 3993:so the equations in characteristic form are: 3590:force in a certain cross section. And, for a 2264:and the surface elevation ζ at each position 1526:), we have (without lateral viscous forces): 8: 1808:one-dimensional (1-D) Saint-Venant equations 5839:-component of the Navier–Stokes equations: 4306:Hamiltonian structure for frictionless flow 1814:, and are commonly used to model transient 7764: 7750: 7742: 6657:, later on. The water depth is 100 meters. 3808:{\displaystyle c={\sqrt {\frac {gA}{B}}}.} 3498: 3366: 3252: 1829:The 1-D equations are used extensively in 7602: 7545: 7223: 7174: 6972: 6970: 6968: 6904:Comptes Rendus de l'Académie des Sciences 6605:Wave modelling by shallow-water equations 6495: 6457: 6424: 6398: 6372: 6370: 6337: 6318: 6312: 6271: 6265: 6225: 6199: 6167: 6153: 6080: 6074: 6018: 5999: 5986: 5980: 5940: 5895: 5885: 5869: 5849: 5844: 5793: 5761: 5718: 5692: 5687: 5646: 5628: 5621: 5609: 5591: 5584: 5572: 5554: 5547: 5537: 5517: 5444: 5423: 5405: 5398: 5386: 5368: 5361: 5349: 5331: 5324: 5303: 5283: 5257: 5231: 5205: 5179: 5177: 5140:Learn how and when to remove this message 5047: 5041: 4982: 4949: 4944: 4817: 4791: 4765: 4727: 4708: 4685: 4631: 4608: 4563: 4540: 4502: 4483: 4460: 4449: 4447: 4405: 4394: 4374: 4365: 4348: 4329: 4270: 4250: 4240: 4237: 4231: 4217: 4181: 4156: 4150: 4148: 4138: 4127: 4107: 4097: 4094: 4088: 4074: 4038: 4013: 4007: 4005: 4000: 3998: 3970: 3952: 3946: 3921: 3903: 3897: 3785: 3777: 3736: 3726: 3723: 3721: 3499: 3458: 3457: 3431: 3426: 3394: 3381: 3367: 3351: 3340: 3314: 3309: 3277: 3253: 3223: 3218: 3185: 3183: 3167:. So σ is the height above the bed level 3092: 3087: 3064: 3054: 3050: 3033: 3028: 3011: 3005: 2985: 2962: 2960: 2860: 2859: 2849: 2823: 2822: 2796: 2795: 2769: 2767: 2726: 2718: 2685: 2675: 2674: 2668: 2640: 2631: 2630: 2621: 2618: 2607: 2447: 2403: 2398: 2371: 2096: 2095: 2085: 2059: 2058: 2032: 2031: 2005: 2003: 1928: 1905: 1903: 1812:Adhémar Jean Claude Barré de Saint-Venant 1753: 1711: 1675: 1633: 1595: 1572: 1537: 1533: 1531: 1208: 1190: 1183: 1171: 1153: 1146: 1106: 1068: 1042: 1012: 991: 973: 966: 954: 936: 929: 889: 851: 825: 795: 776: 775: 751: 750: 735: 726: 725: 701: 700: 685: 658: 654: 652: 538: 524: 504: 495: 465: 426: 378: 364: 344: 335: 305: 266: 221: 186: 150: 146: 144: 50:Adhémar Jean Claude Barré de Saint-Venant 38:hyperbolic partial differential equations 7708:Numerical Methods for Shallow-Water Flow 6745:Numerical Methods for Shallow-Water Flow 2526:, they can be related by using e.g. the 93: 7496:, Advanced Series on Ocean Engineering 6946: 6944: 6734: 5086:Derivation from Navier–Stokes equations 1512:, and is equivalent to saying that the 8036:one-dimensional Saint-Venant equations 3986:{\displaystyle r_{-}=u-2{\sqrt {gh}},} 3164: 2353: 1802:One-dimensional Saint-Venant equations 7397: 7395: 6113:{\displaystyle F_{g}=\sin(\theta )gM} 6047:is the body force due to gravity and 6033:{\displaystyle f_{x}=f_{x,g}+f_{x,f}} 5678:Assuming one-dimensional flow in the 3934:{\displaystyle r_{+}=u+2{\sqrt {gh}}} 2751:Consequently, the momentum equation ( 1504:, which represent the effect of bulk 7: 8983: 7494:Wave propagation over uneven bottoms 7404:Modeling and control of hydrosystems 6863: 6861: 6859: 6857: 2952: 2921:) can also be cast in the so-called 2759: 2226:. Further ρ is the (constant) fluid 1995: 1895: 7583:Journal of the Atmospheric Sciences 7212:Applied Mathematics and Computation 6686:, is that it allows solutions like 2303:. The instantaneous water depth is 139:, the shallow-water equations are: 8863:National Oceanographic Data Center 8290:World Ocean Circulation Experiment 8178:Global Ocean Data Analysis Project 6435: 6427: 6409: 6401: 6383: 6375: 6054:is the body force due to friction. 5951: 5943: 5922: 5909: 5860: 5852: 5813: 5795: 5729: 5721: 5703: 5695: 5639: 5625: 5602: 5588: 5565: 5551: 5416: 5402: 5379: 5365: 5342: 5328: 5294: 5286: 5268: 5260: 5242: 5234: 5216: 5208: 5190: 5182: 4960: 4952: 4828: 4820: 4802: 4794: 4776: 4768: 4738: 4730: 4714: 4710: 4696: 4688: 4651: 4634: 4619: 4611: 4583: 4566: 4551: 4543: 4513: 4505: 4489: 4485: 4471: 4463: 4406: 4251: 4241: 4157: 4151: 4108: 4098: 4014: 4008: 3737: 3727: 3716:on the characteristic curves are: 3531:Above – in the momentum equation ( 3500: 3489: 3461: 3368: 3254: 2991: 2987: 2973: 2965: 2861: 2834: 2826: 2807: 2799: 2780: 2772: 2676: 2641: 2632: 2622: 2522:is dependent on the flow velocity 2070: 2062: 2043: 2035: 2016: 2008: 1952: 1931: 1916: 1908: 1873:Cross section of the open channel. 1764: 1756: 1722: 1714: 1686: 1678: 1644: 1636: 1606: 1598: 1583: 1575: 1548: 1540: 1201: 1187: 1164: 1150: 1117: 1109: 1079: 1071: 1053: 1045: 1023: 1015: 984: 970: 947: 933: 900: 892: 862: 854: 836: 828: 806: 798: 741: 737: 691: 687: 669: 661: 564: 541: 471: 467: 449: 429: 404: 381: 311: 307: 289: 269: 244: 224: 209: 189: 170: 153: 14: 8710:Global Sea Level Observing System 4442:. Hamilton's equations then are: 2742:{\displaystyle R={\frac {A}{P}}.} 8993: 8982: 8973: 8972: 8168:Geochemical Ocean Sections Study 8084: 8073: 7028:Fluid Dynamics/Strömungsmechanik 5172:-direction – can be written as: 5094: 8898:Ocean thermal energy conversion 8621:Vine–Matthews–Morley hypothesis 7522:"Shallow water wave turbulence" 6560: 6541: 6522: 6506: 6352:{\displaystyle f_{x,f}=S_{f}g.} 6127:is the force of gravity in the 4236: 4137: 4093: 3834:determines whether the flow is 3639:is continuous across the jump. 3380: 2279:, the cross sectional area is: 113:conservation of linear momentum 9026:Partial differential equations 7710:, Kluwer Academic Publishers, 7684:, Cambridge University Press, 7682:Unsteady flow in open channels 7006:, Cambridge University Press, 6567: 6561: 6557: 6551: 6538: 6532: 6519: 6513: 6503: 6497: 6469: 6450: 6101: 6095: 5819: 5810: 4994: 4975: 4646: 4637: 4578: 4569: 3484: 3467: 3454: 3437: 3412: 3400: 3363: 3344: 3337: 3320: 3295: 3283: 3249: 3232: 3204: 3192: 2444: 2427: 2419: 2407: 2388: 2376: 2246:hyperbolic system of equations 1879:partial differential equations 1488:of Carrier and Yeh (2005) for 769: 757: 719: 707: 559: 544: 444: 432: 399: 384: 284: 272: 239: 227: 204: 192: 165: 156: 123:. In the case of a horizontal 1: 7577:Bühler, Oliver (1998-09-01). 6900:Saint-Venant, A.J.C. Barré de 6810:"The Shallow Water Equations" 6778:"The Shallow Water Equations" 6682:. An advantage of this, over 3625:), or the conservation form ( 61:Navier–Stokes equations 8158:El Niño–Southern Oscillation 8128:Craik–Leibovich vortex force 7884:Luke's variational principle 7738:, some analytical solutions) 7734:(instead of simplifying the 6983:, §§5.2 & 13.10, Wiley, 5682:-direction it follows that: 3700:) can be analysed using the 3688:The Saint-Venant equations ( 619:) is the fluid's horizontal 9021:Equations of fluid dynamics 7463:. Bentley Institute Press. 7241:Pure and Applied Geophysics 7108:Pure and Applied Geophysics 6868:S. Néelz; G Pender (2009). 6684:Quasi-geostrophic equations 6298:{\displaystyle f_{x,g}=gS.} 5532:can be assumed as zero so: 5120:the claims made and adding 4318:structure. The Hamiltonian 3696: 3690: 3627: 3621: 3615: 3533: 2933: 2927: 2917: 2753: 2553: 2543: 2362: 2256: 2250: 46:(de) Saint-Venant equations 9052: 8223:Ocean dynamical thermostat 8071: 7706:Vreugdenhil, C.B. (1994), 7526:Journal of Fluid Mechanics 7459:Methods., Haestad (2007). 7402:Vincent., Fromion (2009). 7085:10.1088/0951-7715/24/3/R01 7030:, Encyclopedia of Physics 6981:Linear and Nonlinear Waves 6742:Vreugdenhil, C.B. (1986). 5068:{\displaystyle S_{f}-S=0.} 2515:for rectangular channels. 2272:and channel bed elevation 2236:gravitational acceleration 2222:) of the cross section at 1458: 1446: 1412: 1395: 1291: 1281: 1263: 1245: 8968: 8758:Ocean acoustic tomography 8571:Mohorovičić discontinuity 8163:General circulation model 7799:Benjamin–Feir instability 7261:10.1007/s00024-017-1476-3 7225:10.1016/j.amc.2016.01.005 7176:10.1007/s40722-015-0040-4 7128:10.1007/s00024-014-1016-3 6849:10.3970/cmes.2005.010.113 6754:10.1007/978-94-015-8354-1 5788:or in differential form: 5781:{\displaystyle p=\rho gh} 5504:is the body force in the 3702:method of characteristics 3537:) in conservation form – 2155:) is the cross-sectional 1431:is equal to 2Ω sin( 8888:Ocean surface topography 8263:Thermohaline circulation 8253:Subsurface ocean current 8193:Hydrothermal circulation 8026:Wave–current interaction 7804:Boussinesq approximation 7680:; Labeur, R. J. (2017), 7492:Dingemans, M.W. (1997), 7440:www.physicsclassroom.com 6690:, while also conserving 6628:Boussinesq approximation 5020:and LISFLOOD-FP. In the 2911:Conservation of momentum 2488:when the fluid depth is 2159:of the flow at location 8925:Sea surface temperature 8908:Outline of oceanography 8103:Atmospheric circulation 8041:shallow water equations 8031:Waves and shallow water 7924:Significant wave height 7736:Navier–Stokes equations 6928:Open-channel hydraulics 6724:Waves and shallow water 5485:is the velocity in the 5477:is the velocity in the 5469:is the velocity in the 5154:Navier–Stokes equations 2915:The momentum equation ( 2528:Darcy–Weisbach equation 1881:which describe the 1-D 1270:is the velocity in the 1252:is the velocity in the 117:Navier–Stokes equations 30:shallow-water equations 8920:Sea surface microlayer 8285:Wind generated current 6671: 6658: 6577: 6485: 6353: 6299: 6245: 6184: 6146: 6114: 6034: 5967: 5829: 5782: 5748: 5670: 5526: 5457: 5069: 5007: 4859: 4432:the channel width and 4420: 4291: 3987: 3935: 3809: 3766: 3685: 3523: 3111: 2888: 2743: 2707: 2658: 2518:The wall shear stress 2466: 2113: 1971: 1874: 1793: 1493: 1233: 611:), and the 2D vector ( 587: 99: 25: 9031:Physical oceanography 8753:Deep scattering layer 8735:World Geodetic System 8243:Princeton Ocean Model 8123:Coriolis–Stokes force 7773:Physical oceanography 7690:10.1017/9781316576878 6954:, Pitman Publishing, 6910:: 147–154 and 237–240 6669: 6647:Boussinesq-type model 6641: 6578: 6486: 6354: 6300: 6246: 6185: 6144: 6115: 6035: 5968: 5830: 5783: 5749: 5671: 5527: 5458: 5166:Cartesian coordinates 5070: 5008: 4860: 4421: 4292: 3988: 3936: 3810: 3767: 3650: 3524: 3112: 2889: 2757:) can be written as: 2744: 2708: 2659: 2467: 2366:) can be written as: 2114: 1972: 1872: 1824:cross-sectional shape 1794: 1482: 1234: 635:Non-conservative form 588: 97: 82:of atmospheric flow. 22: 8773:Underwater acoustics 8333:Perigean spring tide 8198:Langmuir circulation 7909:Rossby-gravity waves 7547:10.1017/jfm.2019.375 7042:, Springer, Berlin, 6938:, §18-1 & §18-2. 6494: 6369: 6311: 6264: 6198: 6152: 6073: 5979: 5843: 5792: 5760: 5686: 5536: 5525:{\displaystyle \nu } 5516: 5176: 5040: 4943: 4446: 4328: 3997: 3945: 3896: 3776: 3720: 3182: 3165:cross-section figure 2959: 2766: 2717: 2667: 2606: 2370: 2354:cross-section figure 2002: 1902: 1530: 1423:associated with the 1421:Coriolis coefficient 651: 143: 109:conservation of mass 86:describe the state. 8935:Science On a Sphere 8541:Convergent boundary 8213:Modular Ocean Model 8173:Geostrophic current 7889:Mild-slope equation 7644:2017PhFl...29k1114L 7595:1998JAtS...55.2884B 7538:2019JFM...874.1169A 7290:Lannes, D. (2013). 7253:2017PApGe.174.3185A 7120:2015PApGe.172..885H 7077:2011Nonli..24R...1D 6880:on 8 September 2019 6712:potential vorticity 6696:potential vorticity 4438:the constant fluid 3436: 3319: 3228: 2937:). In terms of the 2549:continuity equation 2541:Further, equation ( 2423: 1883:incompressible flow 1510:geostrophic balance 1467:kinematic viscosity 80:primitive equations 8591:Seafloor spreading 8581:Outer trench swell 8546:Divergent boundary 8446:Continental margin 8431:Carbonate platform 8328:Lunitidal interval 7050:, pp. 225–282 6672: 6659: 6573: 6481: 6349: 6295: 6241: 6180: 6147: 6135:is the angle, and 6110: 6030: 5963: 5825: 5778: 5744: 5666: 5522: 5453: 5105:possibly contains 5065: 5003: 4914:2016-10-25 at the 4855: 4853: 4416: 4287: 4285: 3983: 3931: 3890:Riemann invariants 3805: 3762: 3686: 3519: 3517: 3422: 3305: 3214: 3107: 2884: 2739: 2703: 2654: 2602:) are defined as: 2572:), friction slope 2462: 2394: 2109: 1967: 1875: 1789: 1787: 1494: 1229: 1227: 583: 581: 127:, with negligible 100: 26: 9008: 9007: 9000:Oceans portal 8960:World Ocean Atlas 8950:Underwater glider 8893:Ocean temperature 8556:Hydrothermal vent 8521:Submarine volcano 8456:Continental shelf 8436:Coastal geography 8426:Bathymetric chart 8308:Amphidromic point 7996:Wave nonlinearity 7854:Infragravity wave 7699:978-1-107-15029-4 7652:10.1063/1.4985990 7632:Physics of Fluids 7589:(17): 2884–2891. 7507:978-981-02-0427-3 7436:"Inclined Planes" 7048:978-3-642-45946-7 7012:978-0-521-01045-0 6962:, §§2.1 & 2.2 6763:978-90-481-4472-3 6572: 6550: 6547: 6544: 6531: 6528: 6525: 6512: 6509: 6442: 6416: 6390: 6233: 6232: 6229: 6175: 6174: 6171: 5958: 5929: 5893: 5877: 5867: 5736: 5710: 5653: 5616: 5579: 5430: 5393: 5356: 5311: 5301: 5275: 5249: 5223: 5197: 5150: 5149: 5142: 5107:original research 4967: 4885:Derived modelling 4835: 4809: 4783: 4745: 4721: 4703: 4658: 4626: 4590: 4558: 4520: 4496: 4478: 4382: 4356: 4278: 4259: 4234: 4189: 4165: 4141: 4135: 4116: 4091: 4046: 4022: 3978: 3929: 3800: 3799: 3745: 3555:are evaluated at 3496: 3384: 3131: 3130: 3020: 2998: 2980: 2923:conservation form 2908: 2907: 2841: 2814: 2787: 2734: 2701: 2649: 2133: 2132: 2104: 2093: 2077: 2050: 2023: 1991: 1990: 1959: 1923: 1816:open-channel flow 1771: 1729: 1693: 1651: 1613: 1590: 1555: 1480: 1472: 1471: 1215: 1178: 1124: 1086: 1060: 1030: 998: 961: 907: 869: 843: 813: 748: 698: 676: 571: 512: 478: 456: 411: 352: 318: 296: 251: 216: 177: 103:Conservative form 9043: 8998: 8997: 8986: 8985: 8976: 8975: 8915:Pelagic sediment 8853:Marine pollution 8647:Deep ocean water 8516:Submarine canyon 8451:Continental rise 8343:Rule of twelfths 8258:Sverdrup balance 8188:Humboldt Current 8113:Boundary current 8088: 8077: 7894:Radiation stress 7864:Iribarren number 7839:Equatorial waves 7794:Ballantine scale 7789:Airy wave theory 7766: 7759: 7752: 7743: 7720: 7702: 7664: 7663: 7623: 7617: 7616: 7606: 7574: 7568: 7567: 7549: 7517: 7511: 7510: 7489: 7483: 7482: 7456: 7450: 7449: 7447: 7446: 7432: 7426: 7425: 7399: 7390: 7387: 7381: 7378: 7372: 7369: 7363: 7360: 7354: 7347: 7341: 7338: 7332: 7329: 7323: 7320: 7314: 7313: 7287: 7281: 7280: 7236: 7230: 7229: 7227: 7203: 7197: 7196: 7178: 7154: 7148: 7147: 7114:(3–4): 885–899. 7103: 7097: 7096: 7060: 7051: 7021: 7015: 6997: 6991: 6974: 6963: 6948: 6939: 6921: 6912: 6911: 6896: 6890: 6889: 6887: 6885: 6865: 6852: 6851: 6829: 6823: 6822: 6820: 6819: 6814: 6805: 6799: 6798: 6796: 6795: 6789: 6783:. Archived from 6782: 6774: 6768: 6767: 6739: 6582: 6580: 6579: 6574: 6570: 6548: 6545: 6542: 6529: 6526: 6523: 6510: 6507: 6490: 6488: 6487: 6482: 6462: 6461: 6443: 6441: 6433: 6425: 6417: 6415: 6407: 6399: 6391: 6389: 6381: 6373: 6358: 6356: 6355: 6350: 6342: 6341: 6329: 6328: 6304: 6302: 6301: 6296: 6282: 6281: 6250: 6248: 6247: 6242: 6234: 6230: 6227: 6226: 6189: 6187: 6186: 6181: 6176: 6172: 6169: 6168: 6119: 6117: 6116: 6111: 6085: 6084: 6039: 6037: 6036: 6031: 6029: 6028: 6010: 6009: 5991: 5990: 5972: 5970: 5969: 5964: 5959: 5957: 5949: 5941: 5930: 5928: 5920: 5919: 5915: 5896: 5894: 5886: 5878: 5870: 5868: 5866: 5858: 5850: 5834: 5832: 5831: 5826: 5787: 5785: 5784: 5779: 5753: 5751: 5750: 5745: 5737: 5735: 5727: 5719: 5711: 5709: 5701: 5693: 5675: 5673: 5672: 5667: 5659: 5655: 5654: 5652: 5651: 5650: 5637: 5633: 5632: 5622: 5617: 5615: 5614: 5613: 5600: 5596: 5595: 5585: 5580: 5578: 5577: 5576: 5563: 5559: 5558: 5548: 5531: 5529: 5528: 5523: 5462: 5460: 5459: 5454: 5449: 5448: 5436: 5432: 5431: 5429: 5428: 5427: 5414: 5410: 5409: 5399: 5394: 5392: 5391: 5390: 5377: 5373: 5372: 5362: 5357: 5355: 5354: 5353: 5340: 5336: 5335: 5325: 5312: 5304: 5302: 5300: 5292: 5284: 5276: 5274: 5266: 5258: 5250: 5248: 5240: 5232: 5224: 5222: 5214: 5206: 5198: 5196: 5188: 5180: 5145: 5138: 5134: 5131: 5125: 5122:inline citations 5098: 5097: 5090: 5074: 5072: 5071: 5066: 5052: 5051: 5012: 5010: 5009: 5004: 4987: 4986: 4968: 4966: 4958: 4950: 4922:, Wash 123d and 4880: 4864: 4862: 4861: 4856: 4854: 4841: 4837: 4836: 4834: 4826: 4818: 4810: 4808: 4800: 4792: 4784: 4782: 4774: 4766: 4750: 4746: 4744: 4736: 4728: 4722: 4720: 4709: 4704: 4702: 4694: 4686: 4677: 4664: 4660: 4659: 4657: 4649: 4632: 4627: 4625: 4617: 4609: 4596: 4592: 4591: 4589: 4581: 4564: 4559: 4557: 4549: 4541: 4525: 4521: 4519: 4511: 4503: 4497: 4495: 4484: 4479: 4477: 4469: 4461: 4452: 4437: 4431: 4425: 4423: 4422: 4417: 4409: 4404: 4400: 4399: 4398: 4383: 4375: 4370: 4369: 4357: 4349: 4323: 4296: 4294: 4293: 4288: 4286: 4279: 4271: 4260: 4258: 4254: 4248: 4244: 4238: 4235: 4232: 4229: 4227: 4223: 4222: 4221: 4195: 4191: 4190: 4182: 4166: 4164: 4160: 4154: 4149: 4146: 4142: 4139: 4136: 4128: 4117: 4115: 4111: 4105: 4101: 4095: 4092: 4089: 4086: 4084: 4080: 4079: 4078: 4052: 4048: 4047: 4039: 4023: 4021: 4017: 4011: 4006: 4003: 3992: 3990: 3989: 3984: 3979: 3971: 3957: 3956: 3940: 3938: 3937: 3932: 3930: 3922: 3908: 3907: 3887: 3886: 3885: 3872: 3855: 3844: 3833: 3814: 3812: 3811: 3806: 3801: 3795: 3787: 3786: 3771: 3769: 3768: 3763: 3746: 3744: 3740: 3734: 3730: 3724: 3675: 3605: 3585: 3573: 3528: 3526: 3525: 3520: 3518: 3511: 3503: 3497: 3495: 3487: 3477: 3459: 3453: 3435: 3430: 3399: 3398: 3385: 3382: 3379: 3371: 3356: 3355: 3336: 3318: 3313: 3282: 3281: 3265: 3257: 3242: 3227: 3222: 3125: 3116: 3114: 3113: 3108: 3097: 3096: 3080: 3076: 3069: 3068: 3043: 3039: 3038: 3037: 3021: 3016: 3015: 3006: 2999: 2997: 2986: 2981: 2979: 2971: 2963: 2953: 2949: 2902: 2893: 2891: 2890: 2885: 2877: 2873: 2866: 2865: 2864: 2842: 2840: 2832: 2824: 2815: 2813: 2805: 2797: 2788: 2786: 2778: 2770: 2760: 2748: 2746: 2745: 2740: 2735: 2727: 2712: 2710: 2709: 2704: 2702: 2700: 2686: 2681: 2680: 2679: 2663: 2661: 2660: 2655: 2650: 2648: 2644: 2638: 2637: 2636: 2635: 2625: 2619: 2589:hydraulic radius 2514: 2471: 2469: 2468: 2463: 2458: 2443: 2422: 2402: 2336: 2302: 2209:wetted perimeter 2195:elevation and τ( 2127: 2118: 2116: 2115: 2110: 2105: 2097: 2094: 2086: 2078: 2076: 2068: 2060: 2051: 2049: 2041: 2033: 2024: 2022: 2014: 2006: 1996: 1985: 1976: 1974: 1973: 1968: 1960: 1958: 1950: 1949: 1945: 1929: 1924: 1922: 1914: 1906: 1896: 1810:were derived by 1798: 1796: 1795: 1790: 1788: 1772: 1770: 1762: 1754: 1730: 1728: 1720: 1712: 1694: 1692: 1684: 1676: 1652: 1650: 1642: 1634: 1619: 1615: 1614: 1612: 1604: 1596: 1591: 1589: 1581: 1573: 1556: 1554: 1546: 1538: 1525: 1481: 1462: 1443:is the latitude 1392: 1348: 1244: 1243: 1238: 1236: 1235: 1230: 1228: 1221: 1217: 1216: 1214: 1213: 1212: 1199: 1195: 1194: 1184: 1179: 1177: 1176: 1175: 1162: 1158: 1157: 1147: 1125: 1123: 1115: 1107: 1087: 1085: 1077: 1069: 1061: 1059: 1051: 1043: 1031: 1029: 1021: 1013: 1004: 1000: 999: 997: 996: 995: 982: 978: 977: 967: 962: 960: 959: 958: 945: 941: 940: 930: 908: 906: 898: 890: 870: 868: 860: 852: 844: 842: 834: 826: 814: 812: 804: 796: 781: 780: 756: 755: 749: 747: 736: 731: 730: 706: 705: 699: 697: 686: 677: 675: 667: 659: 592: 590: 589: 584: 582: 572: 570: 562: 539: 534: 530: 529: 528: 513: 505: 500: 499: 479: 477: 466: 457: 455: 447: 427: 412: 410: 402: 379: 374: 370: 369: 368: 353: 345: 340: 339: 319: 317: 306: 297: 295: 287: 267: 252: 250: 242: 222: 217: 215: 207: 187: 178: 176: 168: 151: 9051: 9050: 9046: 9045: 9044: 9042: 9041: 9040: 9011: 9010: 9009: 9004: 8992: 8964: 8803: 8777: 8739: 8720:Sea-level curve 8691: 8630: 8616:Transform fault 8566:Mid-ocean ridge 8532: 8525: 8491:Oceanic plateau 8397: 8383:Tidal resonance 8353:Theory of tides 8294: 8203:Longshore drift 8153:Ekman transport 8089: 8083: 8082: 8081: 8080: 8079: 8078: 8069: 8021:Wave turbulence 7954:Trochoidal wave 7879:Longshore drift 7775: 7770: 7728: 7723: 7718: 7705: 7700: 7676: 7672: 7670:Further reading 7667: 7625: 7624: 7620: 7576: 7575: 7571: 7519: 7518: 7514: 7508: 7491: 7490: 7486: 7471: 7458: 7457: 7453: 7444: 7442: 7434: 7433: 7429: 7414: 7401: 7400: 7393: 7388: 7384: 7379: 7375: 7370: 7366: 7361: 7357: 7348: 7344: 7339: 7335: 7330: 7326: 7321: 7317: 7302: 7289: 7288: 7284: 7238: 7237: 7233: 7205: 7204: 7200: 7156: 7155: 7151: 7105: 7104: 7100: 7062: 7061: 7054: 7022: 7018: 7004:Waves in fluids 6998: 6994: 6975: 6966: 6949: 6942: 6930:, McGraw-Hill, 6922: 6915: 6898: 6897: 6893: 6883: 6881: 6867: 6866: 6855: 6831: 6830: 6826: 6817: 6815: 6812: 6807: 6806: 6802: 6793: 6791: 6787: 6780: 6776: 6775: 6771: 6764: 6741: 6740: 6736: 6732: 6720: 6664: 6630:to incorporate 6607: 6492: 6491: 6453: 6434: 6426: 6408: 6400: 6382: 6374: 6367: 6366: 6333: 6314: 6309: 6308: 6267: 6262: 6261: 6259: 6251:and given that 6196: 6195: 6150: 6149: 6126: 6076: 6071: 6070: 6068: 6053: 6046: 6014: 5995: 5982: 5977: 5976: 5950: 5942: 5921: 5908: 5904: 5897: 5859: 5851: 5841: 5840: 5790: 5789: 5758: 5757: 5728: 5720: 5702: 5694: 5684: 5683: 5642: 5638: 5624: 5623: 5605: 5601: 5587: 5586: 5568: 5564: 5550: 5549: 5546: 5542: 5534: 5533: 5514: 5513: 5503: 5440: 5419: 5415: 5401: 5400: 5382: 5378: 5364: 5363: 5345: 5341: 5327: 5326: 5323: 5319: 5293: 5285: 5267: 5259: 5241: 5233: 5215: 5207: 5189: 5181: 5174: 5173: 5146: 5135: 5129: 5126: 5111: 5099: 5095: 5088: 5043: 5038: 5037: 5030: 4978: 4959: 4951: 4941: 4940: 4936: 4916:Wayback Machine 4892: 4887: 4866: 4852: 4851: 4827: 4819: 4801: 4793: 4775: 4767: 4764: 4760: 4737: 4729: 4723: 4713: 4695: 4687: 4675: 4674: 4650: 4633: 4618: 4610: 4607: 4603: 4582: 4565: 4550: 4542: 4536: 4532: 4512: 4504: 4498: 4488: 4470: 4462: 4444: 4443: 4433: 4427: 4390: 4361: 4347: 4343: 4326: 4325: 4319: 4308: 4284: 4283: 4249: 4239: 4228: 4213: 4206: 4202: 4171: 4167: 4155: 4144: 4143: 4106: 4096: 4085: 4070: 4063: 4059: 4028: 4024: 4012: 3995: 3994: 3948: 3943: 3942: 3899: 3894: 3893: 3881: 3879: 3874: 3864: 3850: 3839: 3821: 3788: 3774: 3773: 3735: 3725: 3718: 3717: 3673: 3664: 3652: 3645: 3643:Characteristics 3633:hydraulic jumps 3604: 3595: 3584: 3575: 3556: 3554: 3547: 3516: 3515: 3504: 3488: 3470: 3460: 3446: 3415: 3390: 3387: 3386: 3372: 3347: 3329: 3298: 3273: 3270: 3269: 3258: 3235: 3207: 3180: 3179: 3173: 3150: 3143: 3123: 3088: 3060: 3059: 3055: 3029: 3007: 3004: 3000: 2990: 2972: 2964: 2957: 2956: 2941: 2913: 2900: 2855: 2854: 2850: 2833: 2825: 2806: 2798: 2779: 2771: 2764: 2763: 2715: 2714: 2690: 2670: 2665: 2664: 2639: 2626: 2620: 2604: 2603: 2578: 2532:Manning formula 2493: 2451: 2436: 2368: 2367: 2343: 2330: 2304: 2294: 2280: 2278: 2125: 2069: 2061: 2042: 2034: 2015: 2007: 2000: 1999: 1983: 1951: 1938: 1934: 1930: 1915: 1907: 1900: 1899: 1867: 1831:computer models 1804: 1786: 1785: 1763: 1755: 1731: 1721: 1713: 1708: 1707: 1685: 1677: 1653: 1643: 1635: 1630: 1629: 1605: 1597: 1582: 1574: 1571: 1567: 1557: 1547: 1539: 1528: 1527: 1517: 1474: 1460: 1363: 1303: 1226: 1225: 1204: 1200: 1186: 1185: 1167: 1163: 1149: 1148: 1145: 1141: 1116: 1108: 1078: 1070: 1052: 1044: 1032: 1022: 1014: 1009: 1008: 987: 983: 969: 968: 950: 946: 932: 931: 928: 924: 899: 891: 861: 853: 835: 827: 815: 805: 797: 792: 791: 740: 690: 678: 668: 660: 649: 648: 637: 580: 579: 563: 540: 520: 491: 484: 480: 470: 458: 448: 428: 423: 422: 403: 380: 360: 331: 324: 320: 310: 298: 288: 268: 263: 262: 243: 223: 208: 188: 179: 169: 152: 141: 140: 129:Coriolis forces 105: 92: 76:Coriolis forces 54:related section 36:) are a set of 17: 12: 11: 5: 9049: 9047: 9039: 9038: 9033: 9028: 9023: 9013: 9012: 9006: 9005: 9003: 9002: 8990: 8980: 8969: 8966: 8965: 8963: 8962: 8957: 8952: 8947: 8942: 8940:Stratification 8937: 8932: 8927: 8922: 8917: 8912: 8911: 8910: 8900: 8895: 8890: 8885: 8880: 8875: 8870: 8865: 8860: 8855: 8850: 8845: 8840: 8832: 8830:Color of water 8827: 8825:Benthic lander 8822: 8817: 8811: 8809: 8805: 8804: 8802: 8801: 8796: 8791: 8785: 8783: 8779: 8778: 8776: 8775: 8770: 8765: 8760: 8755: 8749: 8747: 8741: 8740: 8738: 8737: 8732: 8730:Sea level rise 8727: 8725:Sea level drop 8722: 8717: 8712: 8707: 8701: 8699: 8693: 8692: 8690: 8689: 8684: 8679: 8674: 8669: 8664: 8659: 8654: 8649: 8644: 8638: 8636: 8632: 8631: 8629: 8628: 8623: 8618: 8613: 8608: 8603: 8598: 8593: 8588: 8583: 8578: 8573: 8568: 8563: 8561:Marine geology 8558: 8553: 8548: 8543: 8537: 8535: 8527: 8526: 8524: 8523: 8518: 8513: 8508: 8503: 8501:Passive margin 8498: 8496:Oceanic trench 8493: 8488: 8483: 8478: 8473: 8468: 8463: 8458: 8453: 8448: 8443: 8438: 8433: 8428: 8423: 8418: 8413: 8407: 8405: 8399: 8398: 8396: 8395: 8390: 8385: 8380: 8375: 8370: 8365: 8360: 8355: 8350: 8345: 8340: 8335: 8330: 8325: 8320: 8315: 8310: 8304: 8302: 8296: 8295: 8293: 8292: 8287: 8282: 8277: 8272: 8271: 8270: 8260: 8255: 8250: 8245: 8240: 8235: 8230: 8228:Ocean dynamics 8225: 8220: 8215: 8210: 8205: 8200: 8195: 8190: 8185: 8180: 8175: 8170: 8165: 8160: 8155: 8150: 8145: 8140: 8135: 8130: 8125: 8120: 8118:Coriolis force 8115: 8110: 8105: 8099: 8097: 8091: 8090: 8072: 8070: 8068: 8067: 8066: 8065: 8055: 8050: 8045: 8044: 8043: 8038: 8028: 8023: 8018: 8013: 8008: 8003: 7998: 7993: 7988: 7983: 7978: 7973: 7968: 7967: 7966: 7956: 7951: 7946: 7941: 7939:Stokes problem 7936: 7931: 7926: 7921: 7916: 7911: 7906: 7901: 7896: 7891: 7886: 7881: 7876: 7874:Kinematic wave 7871: 7866: 7861: 7856: 7851: 7846: 7841: 7836: 7831: 7826: 7821: 7816: 7811: 7806: 7801: 7796: 7791: 7785: 7783: 7777: 7776: 7771: 7769: 7768: 7761: 7754: 7746: 7740: 7739: 7727: 7726:External links 7724: 7722: 7721: 7717:978-0792331643 7716: 7703: 7698: 7678:Battjes, J. A. 7673: 7671: 7668: 7666: 7665: 7638:(11): 111114. 7618: 7569: 7512: 7506: 7484: 7470:978-0971414167 7469: 7451: 7427: 7412: 7391: 7382: 7373: 7364: 7355: 7342: 7333: 7324: 7315: 7300: 7282: 7231: 7198: 7169:(2): 113–127. 7149: 7098: 7052: 7016: 6992: 6977:Whitham, G. B. 6964: 6940: 6913: 6891: 6853: 6843:(2): 113–122, 6833:Carrier, G. F. 6824: 6800: 6769: 6762: 6733: 6731: 6728: 6727: 6726: 6719: 6716: 6708:kinetic energy 6704:kinetic energy 6663: 6660: 6655:bore formation 6606: 6603: 6590: 6589: 6585: 6584: 6569: 6566: 6563: 6559: 6556: 6553: 6540: 6537: 6534: 6521: 6518: 6515: 6505: 6502: 6499: 6480: 6477: 6474: 6471: 6468: 6465: 6460: 6456: 6452: 6449: 6446: 6440: 6437: 6432: 6429: 6423: 6420: 6414: 6411: 6406: 6403: 6397: 6394: 6388: 6385: 6380: 6377: 6359: 6348: 6345: 6340: 6336: 6332: 6327: 6324: 6321: 6317: 6305: 6294: 6291: 6288: 6285: 6280: 6277: 6274: 6270: 6255: 6240: 6237: 6224: 6221: 6218: 6215: 6212: 6209: 6206: 6203: 6179: 6166: 6163: 6160: 6157: 6124: 6109: 6106: 6103: 6100: 6097: 6094: 6091: 6088: 6083: 6079: 6060: 6055: 6051: 6044: 6027: 6024: 6021: 6017: 6013: 6008: 6005: 6002: 5998: 5994: 5989: 5985: 5973: 5962: 5956: 5953: 5948: 5945: 5939: 5936: 5933: 5927: 5924: 5918: 5914: 5911: 5907: 5903: 5900: 5892: 5889: 5884: 5881: 5876: 5873: 5865: 5862: 5857: 5854: 5848: 5824: 5821: 5818: 5815: 5812: 5809: 5806: 5803: 5800: 5797: 5777: 5774: 5771: 5768: 5765: 5754: 5743: 5740: 5734: 5731: 5726: 5723: 5717: 5714: 5708: 5705: 5700: 5697: 5691: 5676: 5665: 5662: 5658: 5649: 5645: 5641: 5636: 5631: 5627: 5620: 5612: 5608: 5604: 5599: 5594: 5590: 5583: 5575: 5571: 5567: 5562: 5557: 5553: 5545: 5541: 5521: 5501: 5452: 5447: 5443: 5439: 5435: 5426: 5422: 5418: 5413: 5408: 5404: 5397: 5389: 5385: 5381: 5376: 5371: 5367: 5360: 5352: 5348: 5344: 5339: 5334: 5330: 5322: 5318: 5315: 5310: 5307: 5299: 5296: 5291: 5288: 5282: 5279: 5273: 5270: 5265: 5262: 5256: 5253: 5247: 5244: 5239: 5236: 5230: 5227: 5221: 5218: 5213: 5210: 5204: 5201: 5195: 5192: 5187: 5184: 5156:that describe 5148: 5147: 5102: 5100: 5093: 5087: 5084: 5064: 5061: 5058: 5055: 5050: 5046: 5034:kinematic wave 5029: 5028:Kinematic wave 5026: 5002: 4999: 4996: 4993: 4990: 4985: 4981: 4977: 4974: 4971: 4965: 4962: 4957: 4954: 4948: 4935: 4934:Diffusive wave 4932: 4891: 4888: 4886: 4883: 4850: 4847: 4844: 4840: 4833: 4830: 4825: 4822: 4816: 4813: 4807: 4804: 4799: 4796: 4790: 4787: 4781: 4778: 4773: 4770: 4763: 4759: 4756: 4753: 4749: 4743: 4740: 4735: 4732: 4726: 4719: 4716: 4712: 4707: 4701: 4698: 4693: 4690: 4684: 4681: 4678: 4676: 4673: 4670: 4667: 4663: 4656: 4653: 4648: 4645: 4642: 4639: 4636: 4630: 4624: 4621: 4616: 4613: 4606: 4602: 4599: 4595: 4588: 4585: 4580: 4577: 4574: 4571: 4568: 4562: 4556: 4553: 4548: 4545: 4539: 4535: 4531: 4528: 4524: 4518: 4515: 4510: 4507: 4501: 4494: 4491: 4487: 4482: 4476: 4473: 4468: 4465: 4459: 4456: 4453: 4451: 4426:with constant 4415: 4412: 4408: 4403: 4397: 4393: 4389: 4386: 4381: 4378: 4373: 4368: 4364: 4360: 4355: 4352: 4346: 4342: 4339: 4336: 4333: 4307: 4304: 4282: 4277: 4274: 4269: 4266: 4263: 4257: 4253: 4247: 4243: 4230: 4226: 4220: 4216: 4212: 4209: 4205: 4201: 4198: 4194: 4188: 4185: 4180: 4177: 4174: 4170: 4163: 4159: 4153: 4147: 4145: 4134: 4131: 4126: 4123: 4120: 4114: 4110: 4104: 4100: 4087: 4083: 4077: 4073: 4069: 4066: 4062: 4058: 4055: 4051: 4045: 4042: 4037: 4034: 4031: 4027: 4020: 4016: 4010: 4004: 4002: 3982: 3977: 3974: 3969: 3966: 3963: 3960: 3955: 3951: 3928: 3925: 3920: 3917: 3914: 3911: 3906: 3902: 3804: 3798: 3794: 3791: 3784: 3781: 3761: 3758: 3755: 3752: 3749: 3743: 3739: 3733: 3729: 3669: 3660: 3644: 3641: 3602: 3586:describes the 3582: 3552: 3545: 3514: 3510: 3507: 3502: 3494: 3491: 3486: 3483: 3480: 3476: 3473: 3469: 3466: 3463: 3456: 3452: 3449: 3445: 3442: 3439: 3434: 3429: 3425: 3421: 3418: 3416: 3414: 3411: 3408: 3405: 3402: 3397: 3393: 3389: 3388: 3378: 3375: 3370: 3365: 3362: 3359: 3354: 3350: 3346: 3343: 3339: 3335: 3332: 3328: 3325: 3322: 3317: 3312: 3308: 3304: 3301: 3299: 3297: 3294: 3291: 3288: 3285: 3280: 3276: 3272: 3271: 3268: 3264: 3261: 3256: 3251: 3248: 3245: 3241: 3238: 3234: 3231: 3226: 3221: 3217: 3213: 3210: 3208: 3206: 3203: 3200: 3197: 3194: 3191: 3188: 3187: 3171: 3148: 3141: 3129: 3128: 3119: 3117: 3106: 3103: 3100: 3095: 3091: 3086: 3083: 3079: 3075: 3072: 3067: 3063: 3058: 3053: 3049: 3046: 3042: 3036: 3032: 3027: 3024: 3019: 3014: 3010: 3003: 2996: 2993: 2989: 2984: 2978: 2975: 2970: 2967: 2912: 2909: 2906: 2905: 2896: 2894: 2883: 2880: 2876: 2872: 2869: 2863: 2858: 2853: 2848: 2845: 2839: 2836: 2831: 2828: 2821: 2818: 2812: 2809: 2804: 2801: 2794: 2791: 2785: 2782: 2777: 2774: 2738: 2733: 2730: 2725: 2722: 2699: 2696: 2693: 2689: 2684: 2678: 2673: 2653: 2647: 2643: 2634: 2629: 2624: 2617: 2614: 2611: 2576: 2564:The bed slope 2461: 2457: 2454: 2450: 2446: 2442: 2439: 2435: 2432: 2429: 2426: 2421: 2418: 2415: 2412: 2409: 2406: 2401: 2397: 2393: 2390: 2387: 2384: 2381: 2378: 2375: 2341: 2328: 2292: 2276: 2203:) is the wall 2143:denotes time, 2131: 2130: 2121: 2119: 2108: 2103: 2100: 2092: 2089: 2084: 2081: 2075: 2072: 2067: 2064: 2057: 2054: 2048: 2045: 2040: 2037: 2030: 2027: 2021: 2018: 2013: 2010: 1989: 1988: 1979: 1977: 1966: 1963: 1957: 1954: 1948: 1944: 1941: 1937: 1933: 1927: 1921: 1918: 1913: 1910: 1877:The system of 1866: 1863: 1820:surface runoff 1803: 1800: 1784: 1781: 1778: 1775: 1769: 1766: 1761: 1758: 1752: 1749: 1746: 1743: 1740: 1737: 1734: 1732: 1727: 1724: 1719: 1716: 1710: 1709: 1706: 1703: 1700: 1697: 1691: 1688: 1683: 1680: 1674: 1671: 1668: 1665: 1662: 1659: 1656: 1654: 1649: 1646: 1641: 1638: 1632: 1631: 1628: 1625: 1622: 1618: 1611: 1608: 1603: 1600: 1594: 1588: 1585: 1580: 1577: 1570: 1566: 1563: 1560: 1558: 1553: 1550: 1545: 1542: 1536: 1535: 1490:axisymmetrical 1486:exact solution 1470: 1469: 1463: 1457: 1456: 1449: 1445: 1444: 1425:Coriolis force 1417: 1411: 1410: 1400: 1394: 1393: 1356: 1350: 1349: 1296: 1290: 1289: 1286: 1280: 1279: 1274:direction, or 1268: 1262: 1261: 1256:direction, or 1250: 1224: 1220: 1211: 1207: 1203: 1198: 1193: 1189: 1182: 1174: 1170: 1166: 1161: 1156: 1152: 1144: 1140: 1137: 1134: 1131: 1128: 1122: 1119: 1114: 1111: 1105: 1102: 1099: 1096: 1093: 1090: 1084: 1081: 1076: 1073: 1067: 1064: 1058: 1055: 1050: 1047: 1041: 1038: 1035: 1033: 1028: 1025: 1020: 1017: 1011: 1010: 1007: 1003: 994: 990: 986: 981: 976: 972: 965: 957: 953: 949: 944: 939: 935: 927: 923: 920: 917: 914: 911: 905: 902: 897: 894: 888: 885: 882: 879: 876: 873: 867: 864: 859: 856: 850: 847: 841: 838: 833: 830: 824: 821: 818: 816: 811: 808: 803: 800: 794: 793: 790: 787: 784: 779: 774: 771: 768: 765: 762: 759: 754: 746: 743: 739: 734: 729: 724: 721: 718: 715: 712: 709: 704: 696: 693: 689: 684: 681: 679: 674: 671: 666: 663: 657: 656: 645:hydraulic jump 636: 633: 578: 575: 569: 566: 561: 558: 555: 552: 549: 546: 543: 537: 533: 527: 523: 519: 516: 511: 508: 503: 498: 494: 490: 487: 483: 476: 473: 469: 464: 461: 459: 454: 451: 446: 443: 440: 437: 434: 431: 425: 424: 421: 418: 415: 409: 406: 401: 398: 395: 392: 389: 386: 383: 377: 373: 367: 363: 359: 356: 351: 348: 343: 338: 334: 330: 327: 323: 316: 313: 309: 304: 301: 299: 294: 291: 286: 283: 280: 277: 274: 271: 265: 264: 261: 258: 255: 249: 246: 241: 238: 235: 232: 229: 226: 220: 214: 211: 206: 203: 200: 197: 194: 191: 185: 182: 180: 175: 172: 167: 164: 161: 158: 155: 149: 148: 137:viscous forces 121:hydraulic jump 104: 101: 91: 88: 15: 13: 10: 9: 6: 4: 3: 2: 9048: 9037: 9034: 9032: 9029: 9027: 9024: 9022: 9019: 9018: 9016: 9001: 8996: 8991: 8989: 8981: 8979: 8971: 8970: 8967: 8961: 8958: 8956: 8953: 8951: 8948: 8946: 8943: 8941: 8938: 8936: 8933: 8931: 8928: 8926: 8923: 8921: 8918: 8916: 8913: 8909: 8906: 8905: 8904: 8901: 8899: 8896: 8894: 8891: 8889: 8886: 8884: 8881: 8879: 8876: 8874: 8871: 8869: 8866: 8864: 8861: 8859: 8856: 8854: 8851: 8849: 8848:Marine energy 8846: 8844: 8841: 8839: 8838: 8833: 8831: 8828: 8826: 8823: 8821: 8818: 8816: 8815:Acidification 8813: 8812: 8810: 8806: 8800: 8797: 8795: 8792: 8790: 8787: 8786: 8784: 8780: 8774: 8771: 8769: 8768:SOFAR channel 8766: 8764: 8761: 8759: 8756: 8754: 8751: 8750: 8748: 8746: 8742: 8736: 8733: 8731: 8728: 8726: 8723: 8721: 8718: 8716: 8713: 8711: 8708: 8706: 8703: 8702: 8700: 8698: 8694: 8688: 8685: 8683: 8680: 8678: 8675: 8673: 8670: 8668: 8665: 8663: 8660: 8658: 8655: 8653: 8650: 8648: 8645: 8643: 8640: 8639: 8637: 8633: 8627: 8624: 8622: 8619: 8617: 8614: 8612: 8609: 8607: 8604: 8602: 8599: 8597: 8594: 8592: 8589: 8587: 8584: 8582: 8579: 8577: 8576:Oceanic crust 8574: 8572: 8569: 8567: 8564: 8562: 8559: 8557: 8554: 8552: 8551:Fracture zone 8549: 8547: 8544: 8542: 8539: 8538: 8536: 8534: 8528: 8522: 8519: 8517: 8514: 8512: 8509: 8507: 8504: 8502: 8499: 8497: 8494: 8492: 8489: 8487: 8486:Oceanic basin 8484: 8482: 8479: 8477: 8474: 8472: 8469: 8467: 8464: 8462: 8459: 8457: 8454: 8452: 8449: 8447: 8444: 8442: 8439: 8437: 8434: 8432: 8429: 8427: 8424: 8422: 8419: 8417: 8416:Abyssal plain 8414: 8412: 8409: 8408: 8406: 8404: 8400: 8394: 8391: 8389: 8386: 8384: 8381: 8379: 8376: 8374: 8371: 8369: 8366: 8364: 8361: 8359: 8356: 8354: 8351: 8349: 8346: 8344: 8341: 8339: 8336: 8334: 8331: 8329: 8326: 8324: 8323:Internal tide 8321: 8319: 8316: 8314: 8311: 8309: 8306: 8305: 8303: 8301: 8297: 8291: 8288: 8286: 8283: 8281: 8278: 8276: 8273: 8269: 8266: 8265: 8264: 8261: 8259: 8256: 8254: 8251: 8249: 8246: 8244: 8241: 8239: 8236: 8234: 8231: 8229: 8226: 8224: 8221: 8219: 8218:Ocean current 8216: 8214: 8211: 8209: 8206: 8204: 8201: 8199: 8196: 8194: 8191: 8189: 8186: 8184: 8181: 8179: 8176: 8174: 8171: 8169: 8166: 8164: 8161: 8159: 8156: 8154: 8151: 8149: 8146: 8144: 8141: 8139: 8136: 8134: 8131: 8129: 8126: 8124: 8121: 8119: 8116: 8114: 8111: 8109: 8106: 8104: 8101: 8100: 8098: 8096: 8092: 8087: 8076: 8064: 8061: 8060: 8059: 8056: 8054: 8051: 8049: 8046: 8042: 8039: 8037: 8034: 8033: 8032: 8029: 8027: 8024: 8022: 8019: 8017: 8016:Wave shoaling 8014: 8012: 8009: 8007: 8004: 8002: 7999: 7997: 7994: 7992: 7989: 7987: 7984: 7982: 7979: 7977: 7976:Ursell number 7974: 7972: 7969: 7965: 7962: 7961: 7960: 7957: 7955: 7952: 7950: 7947: 7945: 7942: 7940: 7937: 7935: 7932: 7930: 7927: 7925: 7922: 7920: 7917: 7915: 7912: 7910: 7907: 7905: 7902: 7900: 7897: 7895: 7892: 7890: 7887: 7885: 7882: 7880: 7877: 7875: 7872: 7870: 7867: 7865: 7862: 7860: 7859:Internal wave 7857: 7855: 7852: 7850: 7847: 7845: 7842: 7840: 7837: 7835: 7832: 7830: 7827: 7825: 7822: 7820: 7817: 7815: 7812: 7810: 7809:Breaking wave 7807: 7805: 7802: 7800: 7797: 7795: 7792: 7790: 7787: 7786: 7784: 7782: 7778: 7774: 7767: 7762: 7760: 7755: 7753: 7748: 7747: 7744: 7737: 7733: 7730: 7729: 7725: 7719: 7713: 7709: 7704: 7701: 7695: 7691: 7687: 7683: 7679: 7675: 7674: 7669: 7661: 7657: 7653: 7649: 7645: 7641: 7637: 7633: 7629: 7622: 7619: 7614: 7610: 7605: 7600: 7596: 7592: 7588: 7584: 7580: 7573: 7570: 7565: 7561: 7557: 7553: 7548: 7543: 7539: 7535: 7532:: 1169–1196. 7531: 7527: 7523: 7516: 7513: 7509: 7503: 7499: 7495: 7488: 7485: 7480: 7476: 7472: 7466: 7462: 7455: 7452: 7441: 7437: 7431: 7428: 7423: 7419: 7415: 7413:9781848826243 7409: 7405: 7398: 7396: 7392: 7386: 7383: 7377: 7374: 7368: 7365: 7359: 7356: 7352: 7346: 7343: 7337: 7334: 7328: 7325: 7319: 7316: 7311: 7307: 7303: 7301:9780821894705 7297: 7293: 7286: 7283: 7278: 7274: 7270: 7266: 7262: 7258: 7254: 7250: 7246: 7242: 7235: 7232: 7226: 7221: 7217: 7213: 7209: 7202: 7199: 7194: 7190: 7186: 7182: 7177: 7172: 7168: 7164: 7160: 7153: 7150: 7145: 7141: 7137: 7133: 7129: 7125: 7121: 7117: 7113: 7109: 7102: 7099: 7094: 7090: 7086: 7082: 7078: 7074: 7071:(3): R1–R18. 7070: 7066: 7059: 7057: 7053: 7049: 7045: 7041: 7037: 7033: 7029: 7025: 7020: 7017: 7013: 7009: 7005: 7001: 7000:Lighthill, J. 6996: 6993: 6990: 6989:0-471-94090-9 6986: 6982: 6978: 6973: 6971: 6969: 6965: 6961: 6960:0 273 08442 9 6957: 6953: 6947: 6945: 6941: 6937: 6933: 6929: 6925: 6920: 6918: 6914: 6909: 6905: 6901: 6895: 6892: 6879: 6875: 6871: 6864: 6862: 6860: 6858: 6854: 6850: 6846: 6842: 6838: 6834: 6828: 6825: 6811: 6804: 6801: 6790:on 2012-03-16 6786: 6779: 6773: 6770: 6765: 6759: 6755: 6751: 6747: 6746: 6738: 6735: 6729: 6725: 6722: 6721: 6717: 6715: 6713: 6709: 6705: 6701: 6697: 6693: 6689: 6688:gravity waves 6685: 6681: 6677: 6668: 6661: 6656: 6652: 6648: 6644: 6640: 6636: 6633: 6629: 6624: 6620: 6619:gravity waves 6616: 6612: 6604: 6602: 6598: 6596: 6587: 6586: 6564: 6554: 6535: 6516: 6500: 6478: 6475: 6472: 6466: 6463: 6458: 6454: 6447: 6444: 6438: 6430: 6421: 6418: 6412: 6404: 6395: 6392: 6386: 6378: 6364: 6360: 6346: 6343: 6338: 6334: 6330: 6325: 6322: 6319: 6315: 6306: 6292: 6289: 6286: 6283: 6278: 6275: 6272: 6268: 6258: 6254: 6238: 6235: 6222: 6219: 6216: 6213: 6210: 6207: 6204: 6201: 6193: 6177: 6164: 6161: 6158: 6155: 6143: 6139:is the mass. 6138: 6134: 6130: 6123: 6107: 6104: 6098: 6092: 6089: 6086: 6081: 6077: 6067: 6063: 6059: 6056: 6050: 6043: 6025: 6022: 6019: 6015: 6011: 6006: 6003: 6000: 5996: 5992: 5987: 5983: 5974: 5960: 5954: 5946: 5937: 5934: 5931: 5925: 5916: 5912: 5905: 5901: 5898: 5890: 5887: 5882: 5879: 5874: 5871: 5863: 5855: 5846: 5838: 5822: 5816: 5807: 5804: 5801: 5798: 5775: 5772: 5769: 5766: 5763: 5755: 5741: 5738: 5732: 5724: 5715: 5712: 5706: 5698: 5689: 5681: 5677: 5663: 5660: 5656: 5647: 5643: 5634: 5629: 5618: 5610: 5606: 5597: 5592: 5581: 5573: 5569: 5560: 5555: 5543: 5539: 5519: 5511: 5510: 5509: 5507: 5500: 5496: 5492: 5488: 5484: 5480: 5476: 5473:-direction, 5472: 5468: 5463: 5450: 5445: 5441: 5437: 5433: 5424: 5420: 5411: 5406: 5395: 5387: 5383: 5374: 5369: 5358: 5350: 5346: 5337: 5332: 5320: 5316: 5313: 5308: 5305: 5297: 5289: 5280: 5277: 5271: 5263: 5254: 5251: 5245: 5237: 5228: 5225: 5219: 5211: 5202: 5199: 5193: 5185: 5171: 5167: 5163: 5159: 5155: 5144: 5141: 5133: 5123: 5119: 5115: 5109: 5108: 5103:This section 5101: 5092: 5091: 5085: 5083: 5081: 5075: 5062: 5059: 5056: 5053: 5048: 5044: 5035: 5027: 5025: 5023: 5019: 5013: 5000: 4997: 4991: 4988: 4983: 4979: 4972: 4969: 4963: 4955: 4946: 4933: 4931: 4927: 4925: 4921: 4917: 4913: 4910: 4909:InfoWorks_ICM 4906: 4902: 4898: 4889: 4884: 4882: 4878: 4874: 4870: 4848: 4845: 4842: 4838: 4831: 4823: 4814: 4811: 4805: 4797: 4788: 4785: 4779: 4771: 4761: 4757: 4754: 4751: 4747: 4741: 4733: 4724: 4717: 4705: 4699: 4691: 4682: 4679: 4671: 4668: 4665: 4661: 4654: 4643: 4640: 4628: 4622: 4614: 4604: 4600: 4597: 4593: 4586: 4575: 4572: 4560: 4554: 4546: 4537: 4533: 4529: 4526: 4522: 4516: 4508: 4499: 4492: 4480: 4474: 4466: 4457: 4454: 4441: 4436: 4430: 4413: 4410: 4401: 4395: 4391: 4387: 4384: 4379: 4376: 4371: 4366: 4362: 4358: 4353: 4350: 4344: 4340: 4337: 4334: 4331: 4322: 4317: 4313: 4305: 4303: 4300: 4297: 4280: 4275: 4272: 4267: 4264: 4261: 4255: 4245: 4224: 4218: 4214: 4210: 4207: 4203: 4199: 4196: 4192: 4186: 4183: 4178: 4175: 4172: 4168: 4161: 4132: 4129: 4124: 4121: 4118: 4112: 4102: 4081: 4075: 4071: 4067: 4064: 4060: 4056: 4053: 4049: 4043: 4040: 4035: 4032: 4029: 4025: 4018: 3980: 3975: 3972: 3967: 3964: 3961: 3958: 3953: 3949: 3926: 3923: 3918: 3915: 3912: 3909: 3904: 3900: 3891: 3884: 3877: 3871: 3867: 3862: 3857: 3853: 3848: 3847:supercritical 3842: 3837: 3832: 3828: 3824: 3820: 3819:Froude number 3815: 3802: 3796: 3792: 3789: 3782: 3779: 3759: 3756: 3753: 3750: 3747: 3741: 3731: 3715: 3711: 3707: 3703: 3699: 3698: 3693: 3692: 3683: 3679: 3672: 3668: 3663: 3659: 3655: 3649: 3642: 3640: 3638: 3637:momentum flux 3634: 3630: 3629: 3624: 3623: 3618: 3617: 3611: 3609: 3601: 3598: 3593: 3592:non-prismatic 3589: 3581: 3578: 3571: 3567: 3563: 3559: 3551: 3544: 3540: 3536: 3535: 3529: 3512: 3508: 3505: 3492: 3481: 3478: 3474: 3471: 3464: 3450: 3447: 3443: 3440: 3432: 3427: 3423: 3419: 3417: 3409: 3406: 3403: 3395: 3391: 3376: 3373: 3360: 3357: 3348: 3341: 3333: 3330: 3326: 3323: 3315: 3310: 3306: 3302: 3300: 3292: 3289: 3286: 3278: 3274: 3266: 3262: 3259: 3246: 3243: 3239: 3236: 3229: 3224: 3219: 3215: 3211: 3209: 3201: 3198: 3195: 3189: 3177: 3170: 3166: 3162: 3158: 3154: 3147: 3140: 3136: 3127: 3120: 3118: 3104: 3101: 3098: 3093: 3089: 3084: 3081: 3077: 3073: 3070: 3065: 3061: 3056: 3051: 3047: 3044: 3040: 3034: 3030: 3025: 3022: 3017: 3012: 3008: 3001: 2994: 2982: 2976: 2968: 2955: 2954: 2951: 2948: 2944: 2940: 2936: 2935: 2930: 2929: 2924: 2920: 2919: 2910: 2904: 2897: 2895: 2881: 2878: 2874: 2870: 2867: 2856: 2851: 2846: 2843: 2837: 2829: 2819: 2816: 2810: 2802: 2792: 2789: 2783: 2775: 2762: 2761: 2758: 2756: 2755: 2749: 2736: 2731: 2728: 2723: 2720: 2697: 2694: 2691: 2687: 2682: 2671: 2651: 2645: 2627: 2615: 2612: 2609: 2601: 2597: 2593: 2590: 2586: 2582: 2575: 2571: 2567: 2562: 2560: 2556: 2555: 2550: 2546: 2545: 2539: 2537: 2536:Chézy formula 2533: 2529: 2525: 2521: 2516: 2512: 2508: 2504: 2500: 2496: 2491: 2487: 2483: 2479: 2475: 2459: 2455: 2452: 2448: 2440: 2437: 2433: 2430: 2424: 2416: 2413: 2410: 2404: 2399: 2395: 2391: 2385: 2382: 2379: 2373: 2365: 2364: 2360:in equation ( 2359: 2355: 2351: 2347: 2340: 2334: 2327: 2323: 2319: 2315: 2311: 2307: 2301: 2298: 2291: 2287: 2283: 2275: 2271: 2267: 2263: 2259: 2258: 2253: 2252: 2247: 2243: 2239: 2237: 2233: 2229: 2225: 2221: 2217: 2213: 2210: 2206: 2202: 2198: 2194: 2190: 2186: 2182: 2178: 2177:flow velocity 2174: 2170: 2166: 2162: 2158: 2154: 2150: 2146: 2142: 2138: 2129: 2122: 2120: 2106: 2101: 2098: 2090: 2087: 2082: 2079: 2073: 2065: 2055: 2052: 2046: 2038: 2028: 2025: 2019: 2011: 1998: 1997: 1994: 1987: 1980: 1978: 1964: 1961: 1955: 1946: 1942: 1939: 1935: 1925: 1919: 1911: 1898: 1897: 1894: 1892: 1891:cross section 1889:of arbitrary 1888: 1884: 1880: 1871: 1864: 1862: 1860: 1859:flood routing 1856: 1852: 1848: 1844: 1840: 1836: 1832: 1827: 1825: 1821: 1817: 1813: 1809: 1801: 1799: 1782: 1779: 1776: 1773: 1767: 1759: 1750: 1747: 1744: 1741: 1738: 1735: 1733: 1725: 1717: 1704: 1701: 1698: 1695: 1689: 1681: 1672: 1669: 1666: 1663: 1660: 1657: 1655: 1647: 1639: 1626: 1623: 1620: 1616: 1609: 1601: 1592: 1586: 1578: 1568: 1564: 1561: 1559: 1551: 1543: 1524: 1520: 1515: 1514:Rossby number 1511: 1507: 1503: 1499: 1491: 1487: 1468: 1464: 1459: 1454: 1450: 1447: 1442: 1438: 1434: 1430: 1426: 1422: 1418: 1416: 1413: 1409: 1405: 1401: 1399: 1396: 1390: 1386: 1382: 1378: 1374: 1370: 1366: 1361: 1357: 1355: 1352: 1351: 1346: 1342: 1338: 1334: 1330: 1326: 1322: 1318: 1314: 1310: 1306: 1301: 1297: 1295: 1292: 1287: 1285: 1282: 1277: 1273: 1269: 1267: 1264: 1259: 1255: 1251: 1249: 1246: 1242: 1239: 1222: 1218: 1209: 1205: 1196: 1191: 1180: 1172: 1168: 1159: 1154: 1142: 1138: 1135: 1132: 1129: 1126: 1120: 1112: 1103: 1100: 1097: 1094: 1091: 1088: 1082: 1074: 1065: 1062: 1056: 1048: 1039: 1036: 1034: 1026: 1018: 1005: 1001: 992: 988: 979: 974: 963: 955: 951: 942: 937: 925: 921: 918: 915: 912: 909: 903: 895: 886: 883: 880: 877: 874: 871: 865: 857: 848: 845: 839: 831: 822: 819: 817: 809: 801: 788: 785: 782: 772: 766: 763: 760: 744: 732: 722: 716: 713: 710: 694: 682: 680: 672: 664: 646: 642: 634: 632: 630: 626: 622: 621:flow velocity 618: 614: 610: 606: 602: 598: 593: 576: 573: 567: 556: 553: 550: 547: 535: 531: 525: 521: 517: 514: 509: 506: 501: 496: 492: 488: 485: 481: 474: 462: 460: 452: 441: 438: 435: 419: 416: 413: 407: 396: 393: 390: 387: 375: 371: 365: 361: 357: 354: 349: 346: 341: 336: 332: 328: 325: 321: 314: 302: 300: 292: 281: 278: 275: 259: 256: 253: 247: 236: 233: 230: 218: 212: 201: 198: 195: 183: 181: 173: 162: 159: 138: 134: 130: 126: 122: 118: 114: 110: 102: 96: 89: 87: 83: 81: 77: 72: 68: 66: 62: 57: 55: 51: 47: 43: 39: 35: 31: 21: 8955:Water column 8903:Oceanography 8878:Observations 8873:Explorations 8843:Marginal sea 8836: 8794:OSTM/Jason-2 8626:Volcanic arc 8601:Slab suction 8318:Head of tide 8208:Loop Current 8148:Ekman spiral 8040: 7934:Stokes drift 7844:Gravity wave 7819:Cnoidal wave 7707: 7681: 7635: 7631: 7621: 7586: 7582: 7572: 7529: 7525: 7515: 7497: 7493: 7487: 7460: 7454: 7443:. Retrieved 7439: 7430: 7406:. Springer. 7403: 7385: 7376: 7367: 7358: 7350: 7345: 7336: 7327: 7318: 7291: 7285: 7244: 7240: 7234: 7215: 7211: 7201: 7166: 7162: 7152: 7111: 7107: 7101: 7068: 7065:Nonlinearity 7064: 7040:C. Truesdell 7031: 7027: 7024:Meyer, R. E. 7019: 7014:, §§2.8–2.14 7003: 6995: 6980: 6951: 6927: 6924:Chow, Ven Te 6907: 6903: 6894: 6882:. Retrieved 6878:the original 6873: 6840: 6836: 6827: 6816:. Retrieved 6803: 6792:. Retrieved 6785:the original 6772: 6744: 6737: 6673: 6608: 6599: 6591: 6365:-direction: 6362: 6256: 6252: 6191: 6136: 6132: 6131:-direction, 6128: 6121: 6065: 6061: 6057: 6048: 6041: 5836: 5679: 5508:-direction. 5505: 5498: 5494: 5490: 5489:-direction, 5486: 5482: 5481:-direction, 5478: 5474: 5470: 5466: 5464: 5169: 5161: 5158:fluid motion 5151: 5136: 5127: 5104: 5076: 5031: 5022:SIC (Irstea) 5014: 4937: 4928: 4901:SIC (Irstea) 4893: 4890:Dynamic wave 4876: 4872: 4868: 4434: 4428: 4320: 4309: 4301: 4298: 3882: 3875: 3869: 3865: 3863:, i.e. with 3860: 3858: 3851: 3840: 3830: 3826: 3822: 3816: 3713: 3709: 3695: 3689: 3687: 3681: 3677: 3670: 3666: 3661: 3657: 3653: 3626: 3620: 3614: 3612: 3607: 3599: 3596: 3579: 3576: 3569: 3565: 3561: 3557: 3549: 3542: 3538: 3532: 3530: 3175: 3168: 3160: 3156: 3152: 3145: 3138: 3134: 3132: 3121: 2946: 2942: 2932: 2926: 2916: 2914: 2898: 2752: 2750: 2599: 2595: 2591: 2584: 2580: 2573: 2569: 2565: 2563: 2552: 2542: 2540: 2523: 2519: 2517: 2510: 2506: 2502: 2498: 2494: 2489: 2485: 2481: 2477: 2473: 2361: 2357: 2345: 2338: 2332: 2325: 2321: 2317: 2313: 2309: 2305: 2299: 2296: 2289: 2285: 2281: 2273: 2269: 2265: 2261: 2255: 2249: 2240: 2231: 2223: 2219: 2215: 2211: 2205:shear stress 2200: 2196: 2193:free surface 2188: 2184: 2180: 2172: 2168: 2164: 2160: 2152: 2148: 2144: 2140: 2136: 2134: 2123: 1992: 1981: 1887:open channel 1876: 1843:SIC (Irstea) 1828: 1807: 1805: 1522: 1518: 1501: 1497: 1495: 1455:coefficient 1453:viscous drag 1440: 1439:/hour), and 1432: 1428: 1427:. On Earth, 1414: 1404:acceleration 1397: 1388: 1384: 1380: 1376: 1372: 1368: 1364: 1359: 1353: 1344: 1340: 1336: 1332: 1328: 1324: 1320: 1316: 1312: 1308: 1304: 1299: 1293: 1283: 1271: 1265: 1253: 1247: 1240: 641:product rule 638: 624: 616: 612: 608: 604: 600: 596: 594: 106: 84: 73: 69: 58: 45: 42:free surface 33: 29: 27: 9036:Water waves 8945:Thermocline 8662:Mesopelagic 8635:Ocean zones 8606:Slab window 8471:Hydrography 8411:Abyssal fan 8378:Tidal range 8368:Tidal power 8363:Tidal force 8248:Rip current 8183:Gulf Stream 8143:Ekman layer 8133:Downwelling 8108:Baroclinity 8095:Circulation 7991:Wave height 7981:Wave action 7964:megatsunami 7944:Stokes wave 7904:Rossby wave 7869:Kelvin wave 7849:Green's law 7247:(8): 3185. 7218:: 187–197. 6700:shock waves 4316:Hamiltonian 3836:subcritical 3588:hydrostatic 3574:. The term 65:hydrostatic 9015:Categories 8883:Reanalysis 8782:Satellites 8763:Sofar bomb 8611:Subduction 8586:Ridge push 8481:Ocean bank 8461:Contourite 8388:Tide gauge 8373:Tidal race 8358:Tidal bore 8348:Slack tide 8313:Earth tide 8233:Ocean gyre 8053:Wind setup 8048:Wind fetch 8011:Wave setup 8006:Wave radar 8001:Wave power 7899:Rogue wave 7829:Dispersion 7445:2017-05-16 7310:2012046540 6884:2 December 6818:2013-03-28 6794:2010-01-22 6680:turbulence 6676:turbulence 6632:dispersion 6623:wavelength 6190:For small 5130:April 2018 5114:improve it 3706:celerities 3704:. The two 3163:, see the 2352:, see the 2207:along the 1276:meridional 133:frictional 8745:Acoustics 8697:Sea level 8596:Slab pull 8533:tectonics 8441:Cold seep 8403:Landforms 8280:Whirlpool 8275:Upwelling 8058:Wind wave 7986:Wave base 7914:Sea state 7834:Edge wave 7824:Cross sea 7660:1070-6631 7613:0022-4928 7564:198976015 7556:1469-7645 7479:636350249 7422:401159458 7277:132114728 7269:0033-4553 7193:123725815 7185:2198-6444 7136:0033-4553 7036:S. Flügge 6464:− 6436:∂ 6428:∂ 6410:∂ 6402:∂ 6384:∂ 6376:∂ 6220:θ 6217:⁡ 6208:θ 6205:⁡ 6162:θ 6159:⁡ 6099:θ 6093:⁡ 5952:∂ 5944:∂ 5935:− 5923:∂ 5910:∂ 5899:ρ 5891:ρ 5883:− 5875:ρ 5861:∂ 5853:∂ 5847:− 5814:∂ 5805:ρ 5796:∂ 5770:ρ 5730:∂ 5722:∂ 5704:∂ 5696:∂ 5640:∂ 5626:∂ 5603:∂ 5589:∂ 5566:∂ 5552:∂ 5540:ν 5520:ν 5493:is time, 5417:∂ 5403:∂ 5380:∂ 5366:∂ 5343:∂ 5329:∂ 5317:ν 5309:ρ 5295:∂ 5287:∂ 5281:− 5269:∂ 5261:∂ 5243:∂ 5235:∂ 5217:∂ 5209:∂ 5191:∂ 5183:∂ 5118:verifying 5054:− 4989:− 4961:∂ 4953:∂ 4829:∂ 4824:ζ 4821:∂ 4803:∂ 4795:∂ 4777:∂ 4769:∂ 4755:ρ 4742:ζ 4739:∂ 4731:∂ 4715:∂ 4711:∂ 4697:∂ 4689:∂ 4680:ρ 4652:∂ 4635:∂ 4620:∂ 4612:∂ 4601:ρ 4584:∂ 4567:∂ 4552:∂ 4547:ζ 4544:∂ 4530:ρ 4514:∂ 4506:∂ 4490:∂ 4486:∂ 4472:∂ 4467:ζ 4464:∂ 4455:ρ 4392:ζ 4341:∫ 4338:ρ 4312:prismatic 4268:− 4211:− 4176:− 4068:− 3965:− 3954:− 3829:| / 3754:± 3680:and time 3676:in space 3594:channel, 3506:σ 3490:∂ 3472:σ 3462:∂ 3448:σ 3444:− 3441:σ 3433:σ 3424:∫ 3404:σ 3374:σ 3353:′ 3349:σ 3331:σ 3327:− 3324:σ 3316:σ 3307:∫ 3287:σ 3260:σ 3237:σ 3225:σ 3216:∫ 3196:σ 3082:− 3071:− 2992:∂ 2988:∂ 2974:∂ 2966:∂ 2939:discharge 2868:− 2835:∂ 2827:∂ 2808:∂ 2800:∂ 2781:∂ 2773:∂ 2692:ρ 2688:τ 2616:− 2557:) is the 2547:) is the 2396:∫ 2191:) is the 2175:) is the 2102:ρ 2099:τ 2083:− 2071:∂ 2066:ζ 2063:∂ 2044:∂ 2036:∂ 2017:∂ 2009:∂ 1953:∂ 1932:∂ 1917:∂ 1909:∂ 1865:Equations 1774:− 1765:∂ 1757:∂ 1748:− 1723:∂ 1715:∂ 1696:− 1687:∂ 1679:∂ 1670:− 1658:− 1645:∂ 1637:∂ 1607:∂ 1599:∂ 1584:∂ 1576:∂ 1549:∂ 1541:∂ 1506:advection 1278:velocity 1260:velocity 1202:∂ 1188:∂ 1165:∂ 1151:∂ 1139:ν 1127:− 1118:∂ 1110:∂ 1101:− 1080:∂ 1072:∂ 1054:∂ 1046:∂ 1024:∂ 1016:∂ 985:∂ 971:∂ 948:∂ 934:∂ 922:ν 910:− 901:∂ 893:∂ 884:− 872:− 863:∂ 855:∂ 837:∂ 829:∂ 807:∂ 799:∂ 742:∂ 738:∂ 692:∂ 688:∂ 670:∂ 662:∂ 565:∂ 551:η 548:ρ 542:∂ 522:η 515:ρ 489:η 486:ρ 472:∂ 468:∂ 450:∂ 439:η 436:ρ 430:∂ 405:∂ 391:η 388:ρ 382:∂ 362:η 355:ρ 329:η 326:ρ 312:∂ 308:∂ 290:∂ 279:η 276:ρ 270:∂ 245:∂ 234:η 231:ρ 225:∂ 210:∂ 199:η 196:ρ 190:∂ 171:∂ 163:η 160:ρ 154:∂ 90:Equations 52:(see the 8978:Category 8930:Seawater 8657:Littoral 8652:Deep sea 8511:Seamount 8393:Tideline 8338:Rip tide 8268:shutdown 8238:Overflow 7971:Undertow 7814:Clapotis 7144:55004099 7093:59438883 7002:(2005), 6926:(1959), 6718:See also 5032:For the 5018:MIKE SHE 4912:Archived 4897:Mascaret 3825:= | 3509:′ 3475:′ 3451:′ 3377:′ 3334:′ 3263:′ 3240:′ 2559:momentum 2456:′ 2441:′ 1855:MIKE SHE 1839:Mascaret 1833:such as 56:below). 48:, after 8988:Commons 8858:Mooring 8808:Related 8799:Jason-3 8789:Jason-1 8672:Pelagic 8667:Oceanic 8642:Benthic 7959:Tsunami 7929:Soliton 7640:Bibcode 7591:Bibcode 7534:Bibcode 7249:Bibcode 7116:Bibcode 7073:Bibcode 7034:, Eds. 6979:(1974) 6936:4010975 6651:soliton 6643:Tsunami 6595:inertia 5168:in the 5160:. The 5112:Please 5080:HEC-HMS 4920:MIKE 11 4905:HEC-RAS 4899:(EDF), 4440:density 3880:√ 2931:) and ( 2337:, with 2244:of the 2242:Closure 2234:is the 2228:density 1851:MIKE 11 1847:HEC-RAS 1841:(EDF), 1465:is the 1451:is the 1437:radians 1419:is the 1408:gravity 1406:due to 1402:is the 629:density 8677:Photic 8506:Seabed 7919:Seiche 7714: 7696: 7658: 7611: 7562: 7554: 7504: 7477: 7467: 7420: 7410: 7308: 7298: 7275: 7267: 7191: 7183: 7142: 7134: 7091: 7046: 7038:& 7010: 6987: 6958: 6934: 6760: 6692:energy 6615:Kelvin 6611:Rossby 6571: 6549: 6546: 6543: 6530: 6527: 6524: 6511: 6508: 6120:where 6040:where 5465:where 4865:since 3888:, the 3854:> 1 3843:< 1 3619:) or ( 3133:where 2587:) and 2316:) = ζ( 2135:where 1885:in an 1853:, and 1835:TUFLOW 1492:waves. 1241:where 597:η 24:walls. 8868:Ocean 8837:Alvin 8687:Swash 8531:Plate 8476:Knoll 8466:Guyot 8421:Atoll 8300:Tides 8063:model 7949:Swell 7781:Waves 7560:S2CID 7273:S2CID 7189:S2CID 7140:S2CID 7089:S2CID 6813:(PDF) 6788:(PDF) 6781:(PDF) 6730:Notes 6588:Terms 4924:SWMM5 4233:along 4090:along 3892:are: 3845:) or 3772:with 2492:– so 2472:with 2350:datum 2288:(ζ − 1258:zonal 595:Here 115:(the 8835:DSV 8820:Argo 8682:Surf 8138:Eddy 7712:ISBN 7694:ISBN 7656:ISSN 7609:ISSN 7552:ISSN 7502:ISBN 7475:OCLC 7465:ISBN 7418:OCLC 7408:ISBN 7306:LCCN 7296:ISBN 7265:ISSN 7181:ISSN 7132:ISSN 7044:ISBN 7008:ISBN 6985:ISBN 6956:ISBN 6932:OCLC 6886:2016 6758:ISBN 6694:and 6613:and 3941:and 3873:and 3817:The 3548:and 3144:and 2713:and 2505:) = 2324:) − 2295:) = 2230:and 2157:area 1993:and 1818:and 1806:The 1500:and 1375:) = 1331:) + 1319:) = 607:and 135:and 111:and 28:The 7686:doi 7648:doi 7599:doi 7542:doi 7530:874 7257:doi 7245:174 7220:doi 7216:279 7171:doi 7124:doi 7112:172 7081:doi 6845:doi 6750:doi 6231:adj 6228:opp 6214:tan 6202:sin 6173:hyp 6170:opp 6156:sin 6090:sin 6052:x,f 6045:x,g 5116:by 4140:and 3870:B h 3856:). 3694:)–( 3656:= ( 3383:and 3155:(σ, 2534:or 2254:)–( 125:bed 34:SWE 9017:: 7692:, 7654:. 7646:. 7636:29 7634:. 7630:. 7607:. 7597:. 7587:55 7585:. 7581:. 7558:. 7550:. 7540:. 7528:. 7524:. 7498:13 7473:. 7438:. 7416:. 7394:^ 7304:. 7271:. 7263:. 7255:. 7243:. 7214:. 7210:. 7187:. 7179:. 7165:. 7161:. 7138:. 7130:. 7122:. 7110:. 7087:. 7079:. 7069:24 7067:. 7055:^ 7032:IX 6967:^ 6943:^ 6916:^ 6908:73 6906:, 6872:. 6856:^ 6841:10 6839:, 6756:. 6714:. 5664:0. 5082:. 5063:0. 5001:0. 4926:. 4918:, 4907:, 4903:, 4881:. 4875:= 4871:/∂ 3883:gh 3878:= 3868:= 3852:Fr 3841:Fr 3823:Fr 3712:/d 3610:. 3560:= 3541:, 3137:, 2950:: 2947:Au 2945:= 2882:0. 2598:, 2583:, 2538:. 2530:, 2501:, 2284:= 2238:. 2179:, 2163:, 1845:, 1837:, 1826:. 1521:≪ 1379:+ 1371:, 1362:: 1343:, 1339:, 1327:, 1315:, 1311:, 1302:: 603:, 577:0. 131:, 7765:e 7758:t 7751:v 7688:: 7662:. 7650:: 7642:: 7615:. 7601:: 7593:: 7566:. 7544:: 7536:: 7481:. 7448:. 7424:. 7312:. 7279:. 7259:: 7251:: 7228:. 7222:: 7195:. 7173:: 7167:2 7146:. 7126:: 7118:: 7095:. 7083:: 7075:: 6888:. 6847:: 6821:. 6797:. 6766:. 6752:: 6568:) 6565:e 6562:( 6558:) 6555:d 6552:( 6539:) 6536:c 6533:( 6520:) 6517:b 6514:( 6504:) 6501:a 6498:( 6479:, 6476:0 6473:= 6470:) 6467:S 6459:f 6455:S 6451:( 6448:g 6445:+ 6439:x 6431:h 6422:g 6419:+ 6413:x 6405:u 6396:u 6393:+ 6387:t 6379:u 6363:x 6347:. 6344:g 6339:f 6335:S 6331:= 6326:f 6323:, 6320:x 6316:f 6293:. 6290:S 6287:g 6284:= 6279:g 6276:, 6273:x 6269:f 6257:x 6253:f 6239:S 6236:= 6223:= 6211:= 6192:θ 6178:. 6165:= 6137:M 6133:θ 6129:x 6125:g 6122:F 6108:M 6105:g 6102:) 6096:( 6087:= 6082:g 6078:F 6066:g 6064:, 6062:x 6058:f 6049:f 6042:f 6026:f 6023:, 6020:x 6016:f 6012:+ 6007:g 6004:, 6001:x 5997:f 5993:= 5988:x 5984:f 5961:. 5955:x 5947:h 5938:g 5932:= 5926:x 5917:) 5913:h 5906:( 5902:g 5888:1 5880:= 5872:1 5864:x 5856:p 5837:x 5823:. 5820:) 5817:h 5811:( 5808:g 5802:= 5799:p 5776:h 5773:g 5767:= 5764:p 5742:0 5739:= 5733:z 5725:u 5716:w 5713:+ 5707:y 5699:u 5690:v 5680:x 5661:= 5657:) 5648:2 5644:z 5635:u 5630:2 5619:+ 5611:2 5607:y 5598:u 5593:2 5582:+ 5574:2 5570:x 5561:u 5556:2 5544:( 5506:x 5502:x 5499:f 5495:p 5491:t 5487:z 5483:w 5479:y 5475:v 5471:x 5467:u 5451:, 5446:x 5442:f 5438:+ 5434:) 5425:2 5421:z 5412:u 5407:2 5396:+ 5388:2 5384:y 5375:u 5370:2 5359:+ 5351:2 5347:x 5338:u 5333:2 5321:( 5314:+ 5306:1 5298:x 5290:p 5278:= 5272:z 5264:u 5255:w 5252:+ 5246:y 5238:u 5229:v 5226:+ 5220:x 5212:u 5203:u 5200:+ 5194:t 5186:u 5170:x 5162:x 5143:) 5137:( 5132:) 5128:( 5110:. 5060:= 5057:S 5049:f 5045:S 4998:= 4995:) 4992:S 4984:f 4980:S 4976:( 4973:g 4970:+ 4964:x 4956:h 4947:g 4879:) 4877:B 4873:ζ 4869:A 4867:∂ 4849:, 4846:0 4843:= 4839:) 4832:x 4815:g 4812:+ 4806:x 4798:u 4789:u 4786:+ 4780:t 4772:u 4762:( 4758:B 4752:= 4748:) 4734:H 4725:( 4718:x 4706:+ 4700:t 4692:u 4683:B 4672:, 4669:0 4666:= 4662:) 4655:x 4647:) 4644:u 4641:A 4638:( 4629:+ 4623:t 4615:A 4605:( 4598:= 4594:) 4587:x 4579:) 4576:u 4573:A 4570:( 4561:+ 4555:t 4538:B 4534:( 4527:= 4523:) 4517:u 4509:H 4500:( 4493:x 4481:+ 4475:t 4458:B 4435:ρ 4429:B 4414:, 4411:x 4407:d 4402:) 4396:2 4388:B 4385:g 4380:2 4377:1 4372:+ 4367:2 4363:u 4359:A 4354:2 4351:1 4345:( 4335:= 4332:H 4321:H 4281:. 4276:h 4273:g 4265:u 4262:= 4256:t 4252:d 4246:x 4242:d 4225:) 4219:f 4215:S 4208:S 4204:( 4200:g 4197:= 4193:) 4187:h 4184:g 4179:2 4173:u 4169:( 4162:t 4158:d 4152:d 4133:h 4130:g 4125:+ 4122:u 4119:= 4113:t 4109:d 4103:x 4099:d 4082:) 4076:f 4072:S 4065:S 4061:( 4057:g 4054:= 4050:) 4044:h 4041:g 4036:2 4033:+ 4030:u 4026:( 4019:t 4015:d 4009:d 3981:, 3976:h 3973:g 3968:2 3962:u 3959:= 3950:r 3927:h 3924:g 3919:2 3916:+ 3913:u 3910:= 3905:+ 3901:r 3876:c 3866:A 3861:B 3849:( 3838:( 3831:c 3827:u 3803:. 3797:B 3793:A 3790:g 3783:= 3780:c 3760:, 3757:c 3751:u 3748:= 3742:t 3738:d 3732:x 3728:d 3714:t 3710:x 3708:d 3697:2 3691:1 3684:. 3682:t 3678:x 3674:) 3671:P 3667:t 3665:, 3662:P 3658:x 3654:P 3628:4 3622:3 3616:2 3608:x 3603:2 3600:I 3597:g 3583:1 3580:I 3577:g 3572:) 3570:t 3568:, 3566:x 3564:( 3562:h 3558:σ 3553:2 3550:I 3546:1 3543:I 3539:A 3534:4 3513:. 3501:d 3493:x 3485:) 3482:x 3479:, 3468:( 3465:B 3455:) 3438:( 3428:0 3420:= 3413:) 3410:x 3407:, 3401:( 3396:2 3392:I 3369:d 3364:) 3361:x 3358:, 3345:( 3342:B 3338:) 3321:( 3311:0 3303:= 3296:) 3293:x 3290:, 3284:( 3279:1 3275:I 3267:, 3255:d 3250:) 3247:x 3244:, 3233:( 3230:B 3220:0 3212:= 3205:) 3202:x 3199:, 3193:( 3190:A 3176:x 3174:( 3172:b 3169:z 3161:x 3157:x 3153:B 3149:2 3146:I 3142:1 3139:I 3135:A 3126:) 3124:4 3122:( 3105:, 3102:0 3099:= 3094:2 3090:I 3085:g 3078:) 3074:S 3066:f 3062:S 3057:( 3052:A 3048:g 3045:+ 3041:) 3035:1 3031:I 3026:g 3023:+ 3018:A 3013:2 3009:Q 3002:( 2995:x 2983:+ 2977:t 2969:Q 2943:Q 2934:3 2928:1 2918:3 2903:) 2901:3 2899:( 2879:= 2875:) 2871:S 2862:f 2857:S 2852:( 2847:g 2844:+ 2838:x 2830:h 2820:g 2817:+ 2811:x 2803:u 2793:u 2790:+ 2784:t 2776:u 2754:2 2737:. 2732:P 2729:A 2724:= 2721:R 2698:R 2695:g 2683:= 2677:f 2672:S 2652:, 2646:x 2642:d 2633:b 2628:z 2623:d 2613:= 2610:S 2600:t 2596:x 2594:( 2592:R 2585:t 2581:x 2579:( 2577:f 2574:S 2570:x 2568:( 2566:S 2554:2 2544:1 2524:u 2520:τ 2513:) 2511:x 2509:( 2507:B 2503:h 2499:x 2497:( 2495:b 2490:h 2486:x 2482:h 2480:, 2478:x 2476:( 2474:b 2460:, 2453:h 2449:d 2445:) 2438:h 2434:, 2431:x 2428:( 2425:b 2420:) 2417:t 2414:, 2411:x 2408:( 2405:h 2400:0 2392:= 2389:) 2386:t 2383:, 2380:x 2377:( 2374:A 2363:1 2358:A 2346:x 2344:( 2342:b 2339:z 2335:) 2333:x 2331:( 2329:b 2326:z 2322:t 2320:, 2318:x 2314:t 2312:, 2310:x 2308:( 2306:h 2300:h 2297:B 2293:b 2290:z 2286:B 2282:A 2277:b 2274:z 2270:B 2266:x 2262:A 2257:2 2251:1 2248:( 2232:g 2224:x 2220:t 2218:, 2216:x 2214:( 2212:P 2201:t 2199:, 2197:x 2189:t 2187:, 2185:x 2183:( 2181:ζ 2173:t 2171:, 2169:x 2167:( 2165:u 2161:x 2153:t 2151:, 2149:x 2147:( 2145:A 2141:t 2137:x 2128:) 2126:2 2124:( 2107:, 2091:A 2088:P 2080:= 2074:x 2056:g 2053:+ 2047:x 2039:u 2029:u 2026:+ 2020:t 2012:u 1986:) 1984:1 1982:( 1965:0 1962:= 1956:x 1947:) 1943:u 1940:A 1936:( 1926:+ 1920:t 1912:A 1783:. 1780:v 1777:k 1768:y 1760:h 1751:g 1745:= 1742:u 1739:f 1736:+ 1726:t 1718:v 1705:, 1702:u 1699:k 1690:x 1682:h 1673:g 1667:= 1664:v 1661:f 1648:t 1640:u 1627:, 1624:0 1621:= 1617:) 1610:y 1602:v 1593:+ 1587:x 1579:u 1569:( 1565:H 1562:+ 1552:t 1544:h 1523:H 1519:h 1502:v 1498:u 1461:ν 1448:k 1441:φ 1433:φ 1429:f 1415:f 1398:g 1391:) 1389:y 1387:, 1385:x 1383:( 1381:b 1377:D 1373:y 1369:x 1367:( 1365:H 1360:b 1354:b 1347:) 1345:t 1341:y 1337:x 1335:( 1333:h 1329:y 1325:x 1323:( 1321:H 1317:t 1313:y 1309:x 1307:( 1305:η 1300:h 1294:h 1284:H 1272:y 1266:v 1254:x 1248:u 1223:, 1219:) 1210:2 1206:y 1197:v 1192:2 1181:+ 1173:2 1169:x 1160:v 1155:2 1143:( 1136:+ 1133:v 1130:k 1121:y 1113:h 1104:g 1098:= 1095:u 1092:f 1089:+ 1083:y 1075:v 1066:v 1063:+ 1057:x 1049:v 1040:u 1037:+ 1027:t 1019:v 1006:, 1002:) 993:2 989:y 980:u 975:2 964:+ 956:2 952:x 943:u 938:2 926:( 919:+ 916:u 913:k 904:x 896:h 887:g 881:= 878:v 875:f 866:y 858:u 849:v 846:+ 840:x 832:u 823:u 820:+ 810:t 802:u 789:, 786:0 783:= 778:) 773:v 770:) 767:h 764:+ 761:H 758:( 753:( 745:y 733:+ 728:) 723:u 720:) 717:h 714:+ 711:H 708:( 703:( 695:x 683:+ 673:t 665:h 625:g 617:v 615:, 613:u 609:t 605:y 601:x 574:= 568:x 560:) 557:v 554:u 545:( 536:+ 532:) 526:2 518:g 510:2 507:1 502:+ 497:2 493:v 482:( 475:y 463:+ 453:t 445:) 442:v 433:( 420:, 417:0 414:= 408:y 400:) 397:v 394:u 385:( 376:+ 372:) 366:2 358:g 350:2 347:1 342:+ 337:2 333:u 322:( 315:x 303:+ 293:t 285:) 282:u 273:( 260:, 257:0 254:= 248:y 240:) 237:v 228:( 219:+ 213:x 205:) 202:u 193:( 184:+ 174:t 166:) 157:( 32:(

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