Water waves of finite amplitude on a sloping beach

Carrier, G. F.; Greenspan, H. P.

doi:10.1017/s0022112058000331

Cited by 660 publications

(518 citation statements)

References 1 publication

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“…While spurious oscillations at discontinuities are not completely eliminated, they do not grow and result in numerical instabilities. The behaviour of a contact line between wet and dry regions was tested by examining the spreading of a drop of water under gravity (Schär & Smolarkiewicz 1996) and with the solutions of Carrier & Greenspan (1958) for run-up of non-rotating nonlinear shallow water waves on a slope, with very good results in both cases.…”

Section: Discussionmentioning

confidence: 99%

Nonlinear Rossby adjustment in a channel

1999

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The Rossby adjustment problem for a homogeneous fluid in a channel is solved for large values of the initial depth discontinuity. We begin by analysing the classical dam break problem in which the depth on one side of the discontinuity is zero. An approximate solution for this case can be constructed by assuming semigeostrophic dynamics and using the method of characteristics. This theory is supplemented by numerical solutions to the full shallow water equations. The development of the flow and the final, equilibrium volume transport are governed by the ratio of the Rossby radius of deformation to the channel width, the only non-dimensional parameter. After the dam is destroyed the rotating fluid spills down the dry section of the channel forming a rarefying intrusion which, for northern hemisphere rotation, is banked against the right-hand wall (facing downstream). As the channel width is increased the speed of the leading edge (along the right-hand wall) exceeds the intrusion speed for the non-rotating case, reaching the limiting value of 3.80 times the linear Kelvin wave speed in the upstream basin. On the left side of the channel fluid separates from the sidewall at a point whose speed decreases to zero as the channel width approaches infinity. Numerical computations of the evolving flow show good agreement with the semigeostrophic theory for widths less than about a deformation radius. For larger widths cross-channel accelerations, absent in the semigeostrophic approximation, reduce the agreement. The final equilibrium transport down the channel is determined from the semigeostrophic theory and found to depart from the non-rotating result for channels widths greater than about one deformation radius. Rotation limits the transport to a constant maximum value for channel widths greater than about four deformation radii.The case in which the initial fluid depth downstream of the dam is non-zero is then examined numerically. The leading rarefying intrusion is now replaced by a Kelvin shock, or bore, whose speed is substantially less than the zero-depth intrusion speed. The shock is either straight across the channel or attached only to the righthand wall depending on the channel width and the additional parameter, the initial depth difference. The shock speeds and amplitudes on the right-hand wall, for fixed downstream depth, increase above the non-rotating values with increasing channel width. However, rotation reduces the speed of a shock of given amplitude below the non-rotating case. We also find evidence of resonant generation of Poincaré waves by the bore. Shock characteristics are compared to theories of rotating shocks and qualitative agreement is found except for the change in potential vorticity across the shock, which is very sensitive to the model dissipation. Behind the leading shock the flow evolves in much the same way as described by linear theory except for the generation of strongly nonlinear transverse oscillations and rapid advection down the 188 K. R. Helfrich, A. C. Kuo and L. J. Pratt rig...

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Section: Discussionmentioning

confidence: 99%

Nonlinear Rossby adjustment in a channel

1999

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“…This situation has an analytic solution derived by Carrier and Greenspan (1958). Their derivation makes use of the NLSW equations, and thus for consistency the dispersive (l 2 ) terms will be ignored in the numerical simulations for this comparison.…”

Section: Sine Wave Runupmentioning

confidence: 99%

“…The moving boundary technique utilizes linear extrapolation near the wet -dry boundary, thereby allowing the real boundary location to exist in-between nodal points. The model is compared with the classic Carrier and Greenspan (1958) solution for monochromatic long wave runup on a constant slope. As another one horizontal dimension test, the solitary wave runup experiments of Synolakis (1986Synolakis ( , 1987, which range from nonbreaking to breaking waves, are recreated numerically.…”

Section: Introductionmentioning

confidence: 99%

Modeling wave runup with depth-integrated equations

Lynett

Liu

2002

Coastal Engineering

362

212

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In this paper, a moving boundary technique is developed to investigate wave runup and rundown with depth-integrated equations. Highly nonlinear and weakly dispersive equations are solved using a high-order finite difference scheme. An eddy viscosity model is adopted for wave breaking so as to investigate breaking wave runup. The moving boundary technique utilizes linear extrapolation through the wet -dry boundary and into the dry region. Nonbreaking and breaking solitary wave runup is accurately predicted by the proposed model, yielding a validation of both the wave breaking parameterization and the moving boundary technique. Two-dimensional wave runup in a parabolic basin and around a conical island is investigated, and agreement with published data is excellent. Finally, the propagation and runup of a solitary wave in a trapezoidal channel is examined. D

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“…If the slope of the base is sufficiently small, specifically of O(δ), where δ is the aspect ratio of the layer, then, with F = O(1), the flow may be described by the traditional model for shallow water flow on a sloping base (Carrier & Greenspan 1958, Stoker 1957. If, however, the slope of the base exceeds O(δ) then shallow water flow can only be sustained if it is fast enough that the Froude number is large.…”

Section: Uphill Shallow Water Flowsmentioning

confidence: 99%

“…However, the shallow water model with F = 1 may also be used to describe the flow of a thin layer over a gently sloping base, provided that the slope is small enough; a famous example is that of flow on a sloping beach as described in Carrier & Greenspan (1958). When the base slope is of order unity then, as long as F 1, the fluid will flow uphill at least for some distance.…”

Section: Introductionmentioning

confidence: 99%

Non-classical shallow water flows

Edwards¹,

Howison²,

Ockendon³

et al. 2007

IMA Journal of Applied Mathematics

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This paper deals with violent discontinuities in shallow water flows with large Froude number F .On a horizontal base, the paradigm problem is that of the impact of two fluid layers in situations where the flow can be modelled as two smooth regions joined by a singularity in the flow field. Within the framework of shallow water theory we show that, over a certain timescale, this discontinuity may be described by a delta-shock, which is a weak solution of the underlying conservation laws in which the depth and mass and momentum fluxes have both delta function and step function components. We also make some conjectures about how this model evolves from the traditional model for jet impacts in which a spout is emitted.For flows on a sloping base, we show that for flow with an aspect ratio of O(F −2 ) on a base with an O(1) or larger slope, the governing equations admit a new type of discontinuous solution that is also modelled as a deltashock. The physical manifestation of this discontinuity is a small 'tube' of fluid bounding the flow. The delta-shock conditions for this flow are derived and solved for a point source on an inclined plane. This latter delta-shock framework also sheds light on the evolution of the layer impact on a horizontal base.

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Water waves of finite amplitude on a sloping beach

Cited by 660 publications

References 1 publication

Nonlinear Rossby adjustment in a channel

Nonlinear Rossby adjustment in a channel

Modeling wave runup with depth-integrated equations

Non-classical shallow water flows

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