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Dirk Olbers

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Dirk J. Olbers

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A concise theory for scattering of internal waves at localized inhomogeneities (i.e., topographic features, baroclinicity in the density field, variations of the mean sea level, jetlike currents) in the oceanic waveguide is presented within the formal framework of quantum mechanical scattering theory. The equations of motion of the wave system are reduced to a form resembling the Schrödinger equation with an interaction operator describing the effect of the ambient inhomogeneities. By standard Green's function techniques integral equations for the scattered field and its Fourier transform (which relates to the amplitudes of the scattered waves) are derived, both for a scattering region of finite extent (representing a two-dimensional scattering problem) and a “wall-like” scattering region of infinite extent (representing a one-dimensional scattering problem). As an example, the theory is applied to the scattering at a straight geostrophic front. The far-field is described in the Born approximation valid for (U/c)(kL s ) L< 1, where U is the speed of the geostrophic current of width L s , and c and k are the phase speed and wavenumber of the incident wave. It is found that the scattering process has a significant directional signature while modal redistribution appears to be weak.

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Dirk J. Olbers

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The paper presents the WKB theory of internal wave propagation in a large-scale geostrophic mean flow with vertical as well as horizontal shear. As an application a mean flow with isopycnals having constant slope but arbitrary spacing is considered and the behavior of waves at turning points and critical layers is discussed. In particular, it is shown that horizontal variations of the mean flow shift the critical layer to the interior of the wave guide, i.e., away from ω0 2 = f 2, where ω0 is the intrinsic frequency, and produces a valve effect at the critical layer which can be penetrated by a wave incident from one side while incidence from the other side results in absorption.

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Dirk Olbers
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Carsten Eden

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A new type of ocean general circulation model with simplified physics is described and tested for various simple wind-driven circulation problems. The model consists of the vorticity balance of the depth-averaged flow and a hierarchy of equations for “vertical moments” of density and baroclinic velocity. The first vertical density moment is the (vertically integrated) potential energy, which is used to describe the predominant link between the barotropic and the baroclinic oceanic flow in the presence of sloping topography. Tendency equations for the vertical moments of density and baroclinic velocity and an appropriate truncation of the coupled hierarchy of moments are derived that, together with the barotropic vorticity balance, yield a closed set of equations describing the barotropic–baroclinic interaction (BARBI) model of the oceanic circulation. Idealized companion experiments with a numerical implementation of the BARBI model and a primitive equation model indicate that wave propagation properties and baroclinic adjustments are correctly represented in BARBI in midlatitudes as well as in equatorial latitudes. Furthermore, a set of experiments with a realistic application to the Atlantic/Southern Ocean system reproduces important aspects that have been previously reported by studies of gyre circulations and circumpolar currents using full primitive equation models.

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Dirk Olbers
and
Carsten Eden

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Two surface waves can interact to produce an internal gravity wave by nonlinear resonant coupling. The process has been called spontaneous creation (SC) because it operates without internal waves being initially present. Previous studies have shown that the generated internal waves have high frequency close to the local Brunt–Väisälä frequency and wavelengths that are much larger than those of the participating surface waves, and that the spectral transfer rate of energy to the internal wave field is small compared to other generation processes. The aim of the present analysis is to provide a global map of the energy transfer into the internal wave field by surface–internal wave interaction, which is found to be about 10−3 TW in total, based on a realistic wind-sea spectrum (depending on wind speed), mixed layer depths, and stratification below the mixed layer taken from a state-of-the-art numerical ocean model. Unlike previous calculations of the spectral transfer rate based on a vertical mode decomposition, the authors use an analytical framework that directly derives the energy flux of generated internal waves radiating downward from the mixed layer base. Since the radiated waves are of high frequency, they are trapped and dissipated in the upper ocean. The radiative flux thus feeds only a small portion of the water column, unlike in cases of wind-driven near-inertial waves that spread over the entire ocean depth before dissipating. The authors also give an estimate of the interior dissipation and implied vertical diffusivities due to this process. In an extended appendix, they review the modal description of the SC interaction process, completed by the corresponding counterpart, the modulation interaction process (MI), where a preexisting internal wave is modulated by a surface wave and interacts with another one. MI establishes a damping of the internal wave field, thus acting against SC. The authors show that SC overcomes MI for wind speeds exceeding about 10 m s−1.

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Dirk Olbers
and
Martin Visbeck

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The ocean area south of the Antarctic Circumpolar Current (ACC) frontal system is a region of major watermass modification. Influx of North Atlantic Deep Water (NADW), small-scale mixing, eddy transport and diffusion, as well as the fluxes of momentum and buoyancy at the sea surface combine in a complex array of processes to generate the unique stratification of the Southern Ocean with its southward uprising isopycnals and northward flux of Antarctic Intermediate Water (AAIW) and Antarctic Bottom Water. Comprehensive analytical models of this scenario are rare. The authors develop and apply a model based on zonally and temporally averaged theory to explain the conversion of NADW into AAIW with all of the aforementioned processes contained in an extremely simplified way. Eddies appear via a transformed Eulerian mean (TEM) approach with a conventional downgradient parameterization of the meridional density flux. The structure of the eddy coefficient is estimated from hydrographic and wind stress data by a simple inverse approach. Mixing is limited to a near-surface layer and is treated in a most simple entrainment form. The model determines the zonal mean density stratification in the Southern Ocean and the baroclinic transport of the ACC from the applied wind stress and the surface density flux and unravels the role and importance of the different processes responsible for shaping the stratification (Ekman and eddy-induced advection and pumping, mixing, surface buoyancy flux, and eddy-induced diffusion). All of these processes must be present to yield an agreement between the simulated stratification and the observed one, but details of their parameterization might not be too critical. The ACC transport is shown to have a contribution forced by the local wind stress as well as another contribution relating to the nonlocal forcing by wind stress and density flux over the entire Antarctic zone.

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Carsten Eden
and
Dirk Olbers

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The recently proposed Internal Wave Dissipation, Energy and Mixing (IDEMIX) model, describing the propagation and dissipation of internal gravity waves in the ocean, is extended. Compartments describing the energy contained in the internal tides and the near-inertial waves at low, vertical wavenumber are added to a compartment of the wave continuum at higher wavenumbers. Conservation equations for each compartment are derived based on integrated versions of the radiative transfer equation of weakly interacting waves. The compartments interact with each other by the scattering of tidal energy to the wave continuum by triad wave–wave interactions, which are strongly enhanced equatorward of 28° due to parametric subharmonic instability of the tide and by scattering to the continuum of both tidal and near-inertial wave energy over rough topography and at continental margins. Global numerical simulations of the resulting model using observed stratification, forcing functions, and bottom topography yield good agreement with available observations.

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Dirk Olbers
and
Carsten Eden

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When internal (inertia-)gravity waves propagate in a vertically sheared geostrophic (eddying or mean) flow, they exchange energy with the flow. A novel concept parameterizing internal wave–mean flow interaction in ocean circulation models is demonstrated, based on the description of the entire wave field by the wave-energy density in physical and wavenumber space and its prognostic computation by the radiative transfer equation. The concept enables a simplification of the radiative transfer equation with a small number of reasonable assumptions and a derivation of simple but consistent parameterizations in terms of spectrally integrated energy compartments that are used as prognostic model variables. The effect of the waves on the mean flow in this paradigm is in accordance with the nonacceleration theorem: only in the presence of dissipation do waves globally exchange energy with the mean flow in the time mean. The exchange can have either direction. These basic features of wave–mean flow interaction are theoretically derived in a Wentzel–Kramers–Brillouin (WKB) approximation of the wave dynamics and confirmed in a suite of numerical experiments with unidirectional shear flow.

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Carsten Eden
and
Dirk Olbers

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A novel concept for parameterizing internal wave–mean flow interaction in ocean circulation models is extended to an arbitrary two-dimensional flow with vertical shear. The concept is based on the description of the entire wave field by the wave-energy density in physical and wavenumber space and its prognostic computation by the radiative transfer equation integrated in wavenumber space. Energy compartments result for the horizontal direction of wave propagation as additional prognostic model variables, of which only four are taken here for simplicity. The mean flow is interpreted as residual velocities with respect to the wave activity. The effect of wave drag and energy exchange due to the vertical shear of the residual mean flow is then given simply by a vertical flux of momentum. This flux is related to the asymmetries in upward, downward, alongflow, and counterflow wave propagation described by the energy compartments. A numerical implementation in a realistic eddying ocean model shows that the wave drag effect is a significant sink of kinetic energy in the interior ocean.

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Dirk Olbers
and
Carsten Eden

Abstract

An energetically consistent model for the diapycnal diffusivity induced by breaking of internal gravity waves is proposed and tested in local and global settings. The model [Internal Wave Dissipation, Energy and Mixing (IDEMIX)] is based on the spectral radiation balance of the wave field, reduced by integration over the wavenumber space, which yields a set of balances for energy density variables in physical space. A further simplification results in a single partial differential equation for the total energy density of the wave field. The flux of energy to high vertical wavenumbers is parameterized by a functional derived from the wave–wave scattering integral of resonant wave triad interactions, which also forms the basis for estimates of dissipation rates and related diffusivities of ADCP and hydrography fine-structure data. In the current version of IDEMIX, the wave energy is forced by wind-driven near-inertial motions and baroclinic tides, radiating waves from the respective boundary layers at the surface and the bottom into the ocean interior. The model predicts plausible magnitudes and three-dimensional structures of internal wave energy, dissipation rates, and diapycnal diffusivities in rough agreement to observational estimates. IDEMIX is ready for use as a mixing module in ocean circulation models and can be extended with more spectral components.

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