The "shallow-waterness" of the wave climate in European coastal regions

Abstract. In contrast to deep water waves, shallow water waves are influenced by bottom topography, which has consequences for the propagation of wave energy as well as for the energy and momentum exchange between the waves and the mean flow. The ERA-Interim reanalysis is used to assess the fraction of wave energy associated with shallow water waves in coastal regions in Europe. We show maps of the distribution of this fraction as well as time series statistics from 8 selected stations. There is a strong seasonal dependence and high values are typically associated with winter storms, indicating that shallow water wave effects can occasionally be important even in the deeper parts of the shelf seas otherwise dominated by deep water waves.

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The “shallow-waterness” of the wave climate in European coastal regions

Ocean Science ◽

10.5194/os-13-589-2017 ◽

2017 ◽

Vol 13 (4) ◽

pp. 589-597

Author(s):

Kai Håkon Christensen ◽

Ana Carrasco ◽

Jean-Raymond Bidlot ◽

Øyvind Breivik

Keyword(s):

Shallow Water ◽

Deep Water ◽

Wave Energy ◽

Water Waves ◽

Bottom Topography ◽

Mean Flow ◽

Momentum Exchange ◽

Shallow Water Waves ◽

Coastal Regions ◽

Deep Water Waves

Abstract. In contrast to deep water waves, shallow water waves are influenced by bottom topography, which has consequences for the propagation of wave energy as well as for the energy and momentum exchange between the waves and the mean flow. The ERA-Interim reanalysis is used to assess the fraction of wave energy associated with shallow water waves in coastal regions in Europe. We show maps of the distribution of this fraction as well as time series statistics from eight selected stations. There is a strong seasonal dependence and high values are typically associated with winter storms, indicating that shallow water wave effects can occasionally be important even in the deeper parts of the shelf seas otherwise dominated by deep water waves.

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Note on a modification to the nonlinear Schrödinger equation for application to deep water waves

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1979.0154 ◽

1979 ◽

Vol 369 (1736) ◽

pp. 105-114 ◽

Cited By ~ 370

Keyword(s):

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Deep Water ◽

Perturbation Analysis ◽

Water Waves ◽

Schrodinger Equation ◽

Nonlinear Schrodinger Equation ◽

Mean Flow ◽

Nonlinear Schrödinger ◽

Deep Water Waves

The ordinary nonlinear Schrödinger equation for deep water waves, found by perturbation analysis to O (∊ 3 ) in the wave-steepness ∊ ═ ka , is shown to compare rather unfavourably with the exact calculations of Longuet-Higgins (1978 b ) for ∊ > 0.15, say. We show that a significant improvement can be achieved by taking the perturbation analysis one step further O (∊ 4 ). The dominant new effect introduced to order ∊ 4 is the mean flow response to non-uniformities in the radiation stress caused by modulation of a finite amplitude wave.

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On a fourth-order envelope equation for deep-water waves

Journal of Fluid Mechanics ◽

10.1017/s0022112083000014 ◽

1983 ◽

Vol 126 ◽

pp. 1-11 ◽

Cited By ~ 40

Author(s):

Peter A. E. M. Janssen

Keyword(s):

Deep Water ◽

Perturbation Analysis ◽

Water Waves ◽

Mean Flow ◽

Long Time ◽

One Step ◽

Induced Flow ◽

Exact Calculations ◽

Wave Induced ◽

Deep Water Waves

The ordinary nonlinear Schrödinger equation for deep-water waves (found by a perturbation analysis to O(ε3) in the wave steepness ε) compares unfavourably with the exact calculations of Longuet-Higgins (1978) for ε > 0·10. Dysthe (1979) showed that a significant improvement is found by taking the perturbation analysis one step further to O(ε4). One of the dominant new effects is the wave-induced mean flow. We elaborate the Dysthe approach by investigating the effect of the wave-induced flow on the long-time behaviour of the Benjamin–Feir instability. The occurrence of a wave-induced flow may give rise to a Doppler shift in the frequency of the carrier wave and therefore could explain the observed down-shift in experiment (Lake et al. 1977). However, we present arguments why this is not a proper explanation. Finally, we apply the Dysthe equations to a homogeneous random field of gravity waves and obtain the nonlinear energy-transfer function recently found by Dungey & Hui (1979).

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Wave-driven currents and vortex dynamics on barred beaches

Journal of Fluid Mechanics ◽

10.1017/s0022112001006322 ◽

2001 ◽

Vol 449 ◽

pp. 313-339 ◽

Cited By ~ 54

Author(s):

OLIVER BÜHLER ◽

TIVON E. JACOBSON

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Water Waves ◽

Wave Breaking ◽

Bottom Topography ◽

Breaking Waves ◽

Vortex Structures ◽

Water Model ◽

Mean Flow ◽

The Mean

We present a theoretical and numerical investigation of longshore currents driven by breaking waves on beaches, especially barred beaches. The novel feature considered here is that the wave envelope is allowed to vary in the alongshore direction, which leads to the generation of strong dipolar vortex structures where the waves are breaking. The nonlinear evolution of these vortex structures is studied in detail using a simple analytical theory to model the effect of a sloping beach. One of our findings is that the vortex evolution provides a robust mechanism through which the preferred location of the longshore current can move shorewards from the location of wave breaking. Such current dislocation is an often-observed (but ill-understood) phenomenon on real barred beaches.To underpin our results, we present a comprehensive theoretical description of the relevant wave–mean interaction theory in the context of a shallow-water model for the beach. Therein we link the radiation-stress theory of Longuet-Higgins & Stewart to recently established results concerning the mean vorticity generation due to breaking waves. This leads to detailed results for the entire life-cycle of the mean-flow vortex evolution, from its initial generation by wave breaking until its eventual dissipative decay due to bottom friction.In order to test and illustrate our theory we also present idealized nonlinear numerical simulations of both waves and vortices using the full shallow-water equations with bottom topography. In these simulations wave breaking occurs through shock formation of the shallow-water waves. We note that because the shallow-water equations also describe the two-dimensional flow of a homentropic perfect gas, our theoretical and numerical results can also be applied to nonlinear acoustics and sound–vortex interactions.

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