Alastair D. Jenkins

On the Interaction of Surface Waves and Upper Ocean Turbulence

The phase-averaged energy evolution for random surface waves interacting with oceanic turbulence is investigated. The change in wave energy balances the change in the production of turbulent kinetic energy (TKE). Outside the surface viscous layer and the bottom boundary layer the turbulent flux is not related to the wave-induced shear so that eddy viscosity parameterizations cannot be applied. Instead, it is assumed that the wave motion and the turbulent fluxes are not correlated on the scale of the wave period. Using a generalized Lagrangian average it is found that the mean wave-induced shears, despite zero vorticity, yield a production of TKE as if the Stokes drift shear were a mean flow shear. This result provides a new interpretation of a previous derivation from phase-averaged equations by McWilliams et al. It is found that the present source or sink of wave energy is smaller but is still on the order of the empirically adjusted functions used for the dissipation of swell energy in operational wave models, as well as observations of that phenomenon by Snodgrass et al.

Coordinate Transformation on a Sphere Using Conformal Mapping

When setting up global ocean circulation models one faces the problem of including the Arctic Ocean where the traditional spherical coordinate system has a singularity at the pole. In addition, in regional model applications one has to deal with open boundaries where assumptions are made about the normally poorly known boundary conditions. Here an analytical reversible coordinate transformation on a sphere that preserves the orthogonality and the shape of infinitesimal figures is presented. Starting from a standard spherical coordinate system, the transformation is able to map the North and South Poles to two arbitrary locations of the earth and this is readily done with the aid of a conformal mapping in the extended complex plane. The resulting coordinate system will have enhanced resolution along the geodesic curve between the new poles. Examples are given where the transformation is used to strongly increase the resolution in a particular region of interest in the model domain.

Wind and Wave Induced Currents in a Rotating Sea with Depth-varying Eddy Viscosity

Abstract A theory is presented for time-dependent currents induced by a variable wind stress and wave field in deep water away from coastal boundaries. It is based on a second-order perturbation expansion of a version of the Navier-Stokes equations in Lagrangian coordinates. The Coriolis effect and the effect of a depth-dependent eddy viscosity are included. (The eddy viscosity is taken to depend on the Lagrangian vertical coordinate ĉ.) Partial differential equations are derived for the vertical and time variation of the mass transport velocity, together with boundary conditions at the sea surface. The vertical variation of the eddy viscosity causes an extra source term to appear in the equation for the evolution of the current profile. This additional source of momentum within the water column is exactly balanced by an extra term in the surface boundary condition, which in turn represents the contribution to wave dissipation caused by the eddy viscosity within the water column being different from its s...

A Theory for Steady and Variable Wind-and Wave-Induced Currents

Abstract A theory is presented for time-dependent currents induced by a variable wind stress and wave field in deep water away from coastal boundaries. It is based on a second-order perturbation expansion of the Navier-Stokes equations in Lagrangian coordinates. The effects of rotation and of a constant eddy viscosity are included. Partial differential equations are derived for the vertical and time variation of the mass transport velocity, together with boundary conditions at the sea surface. Some simple analytical solutions are presented. For small viscosities, a near-zero mean mass transport is obtained, in agreement with Ursell. Inertial oscillation are superimposed on the above mean solution, in agreement with Hasselmann and Pollard. In the case of a constant wind stress and a constant, horizontally homogeneous wave field, the steady-state results of Weber are reproduced (a surface drift current of about 3% of the wind speed, 23–30 deg to the right of the wind direction).

Comments on “The Three-Dimensional Current and Surface Wave Equations”

Abstract The lowest order sigma-transformed momentum equation given by Mellor takes into account a phase-averaged wave forcing based on Airy wave theory. This equation is shown to be generally inconsistent because of inadequate approximations of the wave motion. Indeed the evaluation of the vertical flux of momentum requires an estimation of the pressure p and coordinate transformation function s to first order in parameters that define the large-scale evolution of the wave field, such as the bottom slope. Unfortunately, there is no analytical expression for p and s at that order. A numerical correction method is thus proposed and verified. Alternative coordinate transforms that allow a separation of wave and mean flow momenta do not suffer from this inconsistency nor do they require a numerical estimation of the wave forcing. Indeed, the problematic vertical flux is part of the wave momentum flux, thus distinct from the mean flow momentum flux, and not directly relevant to the mean flow evolution.

A Quasi-linear Eddy-Viscosity Model for the Flux of Energy and Momentum to Wind Waves Using Conservation-Law Equations in a Curvilinear Coordinate System

Abstract The airflow above ocean waves is calculated using a quasi-linear model—one in which the effect of the waves on the mean flow is taken into account. The model uses curvilinear coordinates, in which one coordinate surface coincides with the instantaneous sea surface, and is consequently able to attain fine vertical resolution in the boundary layer just above the sea surface; the model equations are formulated in conservation-law form. The rates of energy and momentum input to the wave field are calculated from the oscillatory pressure and shear-stress components at the water surface. The equations are solved iteratively using a logarithmically spaced finite-difference mesh. The effect of air turbulence is modeled using a vertically varying shear-stress–dependent eddy viscosity, which acts on the wave-correlated oscillatory motions as well as on the mean flow field. For infinitesimal waves the model agrees with the results of Conte and Miles as the Newtonian viscosity and eddy viscosity that act on ...

Wave damping by a thin layer of viscous fluid

The rate of damping of surface gravity–capillary waves is investigated, in a system which consists of a thin layer of a Newtonian viscous fluid of thickness d floating on a Newtonian fluid of infinite depth. The surface and interfacial tensions, elasticities and viscosities are taken into account. In particular, an approximate dispersion relation is derived for the case where kd and (ω/ν+)1/2d are both small, where k is the wavenumber, ω is the angular frequency and ν+ is the kinematic viscosity of the upper fluid. If d→0 while ν+d remains finite, published dispersion relations for viscoelastic surface films of extremely small (e.g., monomolecular) thickness are reproduced, if we add the surface and interfacial tensions, elasticities and viscosities together, and then add an additional 4ρ+ν+d to the surface viscosity, where ρ+ is the density of the upper fluid. A simple approximation is derived for the damping rate and associated frequency shift when their magnitudes are both small. An example is given of...

Satellite earth observation in operational oceanography

The role and contribution of satellite data in operational oceanography is reviewed, with emphasis on northern European seas. The possibility to observe various ocean parameters and processes by existing satellite sensors, such as optical instruments, infrared radiometers, passive microwave radiometers, and active microwave systems (altimeter, scatterometer, SAR) is discussed. The basic parameters are: sea-surface temperature observed by infrared radiometers, ocean colour by spectrometers, sea-surface elevation by altimeters, and surface roughness by active and passive microwave systems, which can be used to derive surface wind and waves. A number of ocean processes can be derived from synoptic mapping of the basic parameters of larger sea areas, such as current patterns, fronts, eddies, water mass distribution, and various water quality parameters (chlorophyll, surface slicks, suspended sediments). The suitability of existing satellite data to fulfil the operational requirements for temporal and spatial coverage, data delivery in near-real-time, and long-term access to data is discussed in light of the fact that optical/infrared data in northern Europe are severely hampered by frequent cloud cover, while microwave techniques can provide useful data independent of weather and light conditions. Finally, the use of data assimilation in oceanographic models is briefly summarised, indicating that this technique is under development and will soon be adopted in operational oceanography.

Waves and operational oceanography: Toward a coherent description of the upper ocean

The availability of new operational services for ocean circulation modeling presents a unique opportunity to rethink the operational forecasting of ocean waves and how circulation and waves may be combined to provide a better understanding of the upper ocean and enhanced services to society. The largescale oil spill caused by the wreck of the tanker Prestige off the Spanish coast in November 2002, and uncertainties on the fate of that pollution, illustrated the gaps in means of observations and knowledge of relevant processes. The idea of a coupled atmosphere-wavesocean model was proposed by Klaus Hasselmann [Hasselmann, 1991], in the context of climate modeling. As waves are the “gearbox” between the atmosphere and the ocean, a detailed understanding of waves can significantly improve the parameterization of air-sea fluxes and surface processes. Besides, Earth observation systems rely extensively on satellite remote sensing techniques for surface winds, temperature, sea level, ocean color, and sea ice, all affected by surface waves. Hasselmann viewed the future of wave modeling as the development of this central gearbox of a general Earth observation and monitoring system, providing fluxes between ocean and atmosphere in a way consistent with satellite observations. This vision, though slow to materialize, is highly relevant for short-term forecasting in the coastal ocean.

A Lagrangian model for wind- and wave-induced near-surface currents

Abstract A theory is outlined for time-dependent currents induced near the sea surface in deep water, away from coastal boundaries, by a variable wind stress and deep-water wave field. It is based on the theory of Weber (1983) which uses a second-order perturbation expansion of the Navier-Stokes equations in Lagrangian coordinates and includes the Coriolis effect. It uses an eddy viscosity formulation for both wave dissipation and momentum transfer within the current field: the eddy viscosity ν may be allowed to vary with depth. The wind stress may be time-varying and the wave field may vary in both space and time. For the case of a constant ν, the results agree with those of Ursell (1950), Hasselmann (1970) and Pollard (1970) in the limit ν→0, and the steady-state results agree with those of Weber. For a particular case of depth-varying ν, results (obtained from numerical simulations) are in better general agreement with observations of wind-induced surface drift than when a constant ν is used. An outline is given of the application of the theory to the case of a random sea state. There are good prospects for using output data from numerical wave prediction models to drive the equations of this near-surface current model.