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Jan Erik Weber

Abstract

Surface gravity waves in a viscous rotating ocean are studied theoretically when they penetrate an area covered by highly concentrated brashlike ice. The motion is described by a Lagrangian formulation, and the brash is modeled by a viscous Newtonian fluid. Results for wave attenuation and wave drift are obtained in the asymptotic limit of a thin, very viscous upper layer. The derived damping rate compares favorably with field data from the marginal ice zone (MIZ). The drift velocity in the ocean exhibits a marked maximum in the viscous boundary layer near the ice-ocean interface. At the outer edge of the boundary layer it exceeds the inviscid Stokes drift by a factor of 7/4. Computed values for the mean viscous drag on the ice induced by the wave motion show that this effect may compete with the frictional effected of the wind in packing the ice. Finally it is demonstrated that the integrated horizontal mass transports in the open ocean and under the ice do not match, which leads to upwelling in the vicinity of the ice edge.

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Jan Erik H. Weber

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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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Jan Erik Weber

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The steady Ekman boundary-layer current is studied theoretically for the case when the eddy viscosity is proportional to the shear of the wave orbital velocity in a turbulent wave, times the square of a mixing length (Kitaigorodsky, 1961). Assuming a fully developed sea, the wave characteristics, and hence the eddy viscosity distribution with depth, are determined by the wind. The momentum equation is solved numerically to yield the Ekman current as a function of the wind speed. The results show that the magnitude of the Ekman surface current lies between 2.1 and 3% of the wind speed at 10 m height. The deflection angle away from the wind direction is a monotonic decreasing function of wind speed. It varies from 36 to 25° for winds between 5 and 30 m s−1.

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Jan Erik Weber

Abstract

Steady wind-drift currents in a deep viscous rotating ocean are studied theoretically. The analysis is based on the Lagrangian description of motion.

A mean wind-stress at the surface yields the traditional Ekman current. In addition, the wind-stress is assumed to contain a fluctuating part which transfers energy to the surface waves and compensates for loss due to viscous dissipation. The induced drift due to such waves is investigated. The wave-drift depends on the eddy viscosity as well as the earth's rotation.

We assume a fully developed sea, and take the eddy viscosity to be proportional to the friction velocity times a characteristic depth. Hence the total current (Ekman current plus wave-induced current) can be expressed as functions of the wind speed. The results show that the magnitude of the total surface current lies between 3.1 and 3.4% of the wind speed at 10 m height for winds between 5 and 30 m s−1. The deflection angle away from the wind direction varies from 23 to 30° in this range of wind speeds.

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Jan Erik Weber

Abstract

Induced streaming due to deep water surface gravity waves propagating at oblique angles to each other is studied theoretically on the basis of a Lagrangian description. The ocean is slightly viscous, and the primary waves are maintained at constant amplitude by a suitably adjusted small wind stress distribution at the surface. The induced secondary motion in the nonrotating case consists of parallel rolls with axes aligned along the wave propagation direction, and a horizontally undulating Stokes drift. Surface convergence in the roll motion occurs at lines through nodal points of the primary wave system, and downwelling occurs below them. The surface value of the undulating Stokes drift has a minimum at these nodal points if the angle between the crossing waves is less than 76.4°. If this angle is larger than 76.4°, the Stokes drift at the surface has a maximum here. The roll motion described in the present paper is discussed in connection with the basis for the recent theoretical development of Langmuir circulations. Finally, a solution for the steady, horizontally averaged drift current in a rotating ocean is presented.

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Jan Erik Weber

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The mean mass transport induced by surface gravity waves is investigated theoretically for a deep, rotating ocean with a constant eddy viscosity. The waves are periodic in time and have amplitudes that grow or decay slowly in space. The analysis is based on a Lagrangian description of motion, and the results are valid to second order in the wave steepness. An equation for the wave-induced mean Lagrangian mass transport in the oceanic surface layer is derived. It is demonstrated that there are two sources for the mean mass transport: (i) the form drag associated with the fluctuating wind stress normal to the wave slope and (ii) the horizontal divergence of the mean wave momentum flux. Using a spectral formulation for the wave amplitude, applications related to general ocean circulation models are discussed.

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Jan Erik Weber and Arne Melsom

Abstract

A theoretical nonlinear model for wind- and wave-induced currents in a viscous, rotating ocean is developed. The analysis is based on a Lagrangian description of motion. The nonlinear drift problem is formulated such that the solution depends on a linearly increasing eddy viscosity in the water, the wave-growth rate, and the periodic normal (or tangential) wind stress at the sea surface. Particular calculations are performed for surface-stress distributions and growth rates obtained from asymptotic analysis of turbulent atmospheric flow, where the Reynolds stress is modeled by an eddy-viscosity assumption. For growing waves the wave-induced current develops in time. The calculations are terminated when the steepness of the fastest-growing waves reach that of a saturated sea. At this point, the magnitude of the wave-induced surface current is 8–9 times larger than the friction velocity in the water, and the direction of the current is deflected 2°–10° to the right of the wave-propagation direction.

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Jan Erik Weber and Even Førland

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The drift velocity due to capillary-gravity waves in a deep ocean is investigated theoretically. The surface is covered by an insoluble, inextensible film, and the analysis is based an a Lagrangian description of motion. Attenuated as well as nondecaying, or permanent warm, are discussed. The strong temporal attenuation due to the inextensible film is shown to have a profound influence on the drift problem. It causes the induced mean virtual wave stress at the surface to decay quite rapidly, thereby limiting strongly the Growth of the Eulerian part of the drift current. The drift problem for permanent waves is demonstrated to fall into two different categories, depending on the boundary conditions to second order (i) If the mean tangential wind stress vanishes, our results confirm Craik's criticism of the analyses by Phillips and (ii) if tie mean horizontal wind stress vanishes, there is an increased shear in the viscous boundary layer at the surface, as suggested by Phillips. Below the boundary layer the mean flow is essentially dot of Stokes. But the surface velocity, and hence the motion of the film, is in the opposite direction of the wave. Finally, for temporally attenuated waves, it is demonstrated that the difference between the mean drift velocity at a clean surface and the mean drift velocity at a film-covered surface depends very much on the wavelength.

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Frode Høydalsvik and Jan Erik Weber

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The mass transport velocity induced by long surface waves in a shallow, rotating viscous ocean is studied theoretically by using a Lagrangian description of motion. The depth is constant, and the water is homogeneous. Such waves are referred to as Poincaré waves, or sometimes Sverdrup waves, where the latter name usually is reserved for cases in which the effect of friction is taken into account. In the linear case, the primary wave field is significantly affected by the earth's rotation, requiring wave frequencies that are larger than the inertial frequency. In the nonlinear case, the inviscid version of these waves does not induce any mean mass transport. This situation changes when the effect of viscosity is taken into account, and it is shown that for long waves there exists a mean Lagrangian flow confined to a suitably defined bottom friction layer. A solution for constant eddy viscosity and a no-slip bottom is obtained analytically. This result is compared with those obtained numerically for the case in which the eddy viscosity in the bottom layer varies in the vertical and for the case when sliding is allowed at the seabed. In a qualitative sense, the results for the wave drift are surprisingly similar. For waves of the semidiurnal type, it is found that mean drift near the seabed is directed opposite to the wave propagation direction. Possible consequences for the transport of suspended bottom sediments are pointed out.

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Peygham Ghaffari and Jan Erik H. Weber

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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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