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Jan Erik H. Weber

Abstract

The Eulerian volume transport in internal equatorial Kelvin waves subject to viscous attenuation is investigated theoretically by integrating the horizontal momentum equations in the vertical. In terms of small perturbations, the time-averaged horizontal transports are determined to second order in wave steepness. The total Lagrangian volume transport in this problem consists of a Stokes transport plus an Eulerian transport. It is known that the Stokes transport, that is, the vertically integrated Stokes drift, in inviscid internal equatorial Kelvin waves vanishes identically in the rigid-lid approximation for arbitrary vertical variation of the Brunt–Väisälä frequency. The present study considers spatial wave damping due to viscosity. The Stokes transport still becomes zero, but now the radiation stresses due to decaying waves become source terms for the Eulerian mean transport. Calculations of the wave-induced Eulerian transport yield a jetlike symmetric mean flow along the equator from west to east for each baroclinic component, with compensating westward flows on both sides. The flow system scales as the internal equatorial Rossby radius in the north–south direction. The total eastward part of the Eulerian volume flux centered about the equator is estimated to about 0.2 Sv (1 Sv ≡ 106 m3 s−1) for the first baroclinic mode.

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Peygham Ghaffari and Jan Erik H. Weber

Abstract

The Lagrangian mass transport in the Stokes surface edge wave is obtained from the vertically integrated equations of momentum and mass in a viscous rotating ocean, correct to the second order in wave steepness. The analysis is valid for bottom slope angles β in the interval 0 < βπ/2. Vertically averaged drift currents are obtained by dividing the fluxes by the local depth. The Lagrangian mean current is composed of a Stokes drift (inherent in the waves) plus a mean Eulerian drift current. The latter arises as a balance between the radiation stresses, the Coriolis force, and bottom friction. Analytical solutions for the mean Eulerian current are obtained in the form of exponential integrals. The relative importance of the Stokes drift to the Eulerian current in their contribution to the Lagrangian drift velocity is investigated in detail. For the given wavelength, the Eulerian current dominates for medium and large values of β, while for moderate and small β, the Stokes drift yields the main contribution to the Lagrangian drift. Because most natural beaches are characterized by moderate or small slopes, one may only calculate the Stokes drift in order to assess the mean drift of pollution and suspended material in the Stokes edge wave. The main future application of the results for large β appears to be for comparison with laboratory experiments in rotating tanks.

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Jan Erik H. Weber, Kai H. Christensen, and Göran Broström

Abstract

The Stokes drift in long internal equatorial Kelvin waves is investigated theoretically for an inviscid fluid of constant depth. While the Stokes drift in irrotational waves is positive everywhere in the fluid, that is, directed along the phase velocity, this is not always the case for internal Kelvin waves, which possess vorticity. For constant Brunt–Väisälä frequency, the Stokes drift in such waves is sinusoidal in the vertical with a negative value in the middle of the layer for the first baroclinic mode. For a pycnocline that is typical of the equatorial Pacific, this study finds for the first mode that the largest negative Stokes drift velocity occurs near the depth where the Brunt–Väisälä frequency has its maximum. Here, estimated drift values are found to be on the same order of magnitude as those observed in the Pacific Equatorial Undercurrent at the same level. In contrast, a two-layer model with constant density in each layer yields a positive Stokes drift in both layers. This contradicts the fact that, as shown in this paper, the vertically integrated Stokes drift (the Stokes flux) must be zero for arbitrary Brunt–Väisälä frequency.

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Göran Broström, Kai Håkon Christensen, and Jan Erik H. Weber

Abstract

In this study the influence of surface waves on the mean flow in an ocean of arbitrary depth is examined. The wave-induced forcing on the mean flow is obtained by integrating the Eulerian equations for mass and momentum balance from the bottom to an undulating material surface within the water column. By using the mean position of the material surface as the vertical coordinate, the authors obtain the depth dependence of the mean flow and the wave-induced forcing. Substitution of the vertical coordinate makes the model Lagrangian in the vertical direction. The model is Eulerian in the horizontal direction, allowing one to model the effects of a spatially nonuniform wave field or varying depth in a straightforward way.

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Jan Erik H. Weber, Göran Broström, and Øyvind Saetra

Abstract

It is demonstrated that the Eulerian and the Lagrangian descriptions of fluid motion yield the same form for the mean wave-induced volume fluxes in the surface layer of a viscous rotating ocean. In the Eulerian case, the volume fluxes are obtained in the familiar way by integrating the horizontal components of the Navier–Stokes equation in the vertical direction, as seen, for example, in the book by Phillips. In the direct Lagrangian approach, the perturbation equations for the second-order mean drift are integrated in the vertical direction. This yields the advantage that the form drag, which is a source term for the wave-induced transports, can be related to the virtual wave stress that acts to transfer dissipated mean wave momentum into mean currents. In particular, for waves that are periodic in space and time, comparisons between empirical and theoretical relations for the form drag yield an estimate for the wave-induced bulk turbulent eddy viscosity in the surface layer. A simplistic approach extends this analysis to account for wave breaking. By a generalization from a wave component to a wave spectrum, a set of equations for the wave-induced transport in the surface layer is derived for a fully developed sea. Solutions are discussed for an idealized spectral formulation. The problem is formulated such that a numerical wave prediction model can be used to generate the wave-forcing terms in a numerical barotropic ocean surge model. Results from the numerical simulations with a wave-influenced surge model are discussed and compared with similar results from forcing the surge model only by the traditional mean horizontal wind stress computed from the 10-m wind speed. For the simulations presented here, the wave-induced stress constitutes about 50% of the total atmospheric stress for moderate to strong winds.

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Magnus Drivdal, Jan Erik H. Weber, and Jens Boldingh Debernard

Abstract

The dispersion relation for continental shelf waves (CSWs) in a shelf region with an unbounded flat outer ocean, a convex-upward exponential shelf, and an interior flat region of arbitrary width D is derived. The calculations allow for a nonzero divergence of the wave motion. Some consequences of these findings are discussed for the shelf west of Norway, where the shelf at mid-Norway is quite wide, while at Lofoten it is much narrower. Furthermore, north of Lofoten, along the opening to the Barents Sea, the shelf becomes extremely wide. In this region the conditions are nearly met for a double Kelvin wave. The paper discusses how CSWs generated along the common storm track outside southwest Norway modify as they travel along the shelf. The analytical results are compared with results from a numerical barotropic ocean model with realistic shelf topography.

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Peygham Ghaffari, Jan Erik H. Weber, Ole Anders Nøst, and Magnus Drivdal

Abstract

The effect of the continental shelf wave on the flow field over the southern shelf of the Caspian Sea (CS) as the largest enclosed basin of the world, is investigated. Considerable currents with subinertial time scales are observed over the continental shelf in the southern CS. For variations in the surface layer with typical periods of 1 day, local episodic wind events appear to be the driving force. For longer time scales, it is suggested that the observed currents are due to passing continental shelf waves. Measurements over the continental shelf and shelf slope, showing periods of 2–6 days, indicate the presence of such waves. Combined with theory and numerical modeling, the amplitude of the continental shelf wave modes at the coast is assessed from current meter observations. It is demonstrated that the mean drift velocity (the Stokes drift) for long continental shelf waves is determined entirely by the shelf geometry. For the actual shelf mode, it is shown that the associated Stokes drift constitute a nonnegligible mean current along the shelf. This current should be taken into account when assessing the transport of biological material and neutral tracers along the southern coast of the CS.

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