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Carsten Eden

Abstract

In the generalized temporal residual mean (TRM-G) framework, the diapycnal rotational eddy fluxes are defined such that the residual divergent diapycnal eddy flux is related to irreversible changes of buoyancy, that is, diapycnal mixing (or temporal changes of variance and higher order moments) only. Here, it is discussed that for the isopycnal eddy fluxes a similar physically meaningful property exists: rotational isopycnal eddy fluxes can be defined in TRM-G such that the residual divergent part of the flux is related to removal of mean available potential energy and transfer to eddy energy only, that is, to the classical picture of eddy activity. In two idealized eddying models, both featuring strong mesoscale eddy-driven zonal jets, large isopycnal eddy fluxes are circulating at the flanks of the jets. The residual isopycnal eddy fluxes, however, are predominantly meridional and thus downgradient, indicating vanishing anisotropic mixing of isopycnal thickness, consistent with the classical picture of eddy-driven overturning by baroclinic instability in jets. Using isotropic thickness mixing—standard in ocean models—appears therefore as sufficient in this model diagnosis.

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Carsten Eden

Abstract

Three alternative methods of averaging the general conservation equation of a fluid property in a turbulent flow in the Boussinesq approximation are compared: Lagrangian, residual, and isopycnal (or semi-Lagrangian) mean. All methods differentiate consistently but in different ways between effects of advection and irreversible changes of the average property. Because the three average properties differ, the mean transport velocities and the mean irreversible changes in the mean conservation equation differ in general.

The Lagrangian and the semi-Lagrangian (or isopycnal) mean frameworks are shown to be approximately equivalent only for weak irreversible changes, small amplitudes of the turbulent fluctuations, and particle excursion predominantly along the mean property gradient. In that case, the divergent Stokes velocity of the Lagrangian mean framework can be replaced in the Lagrangian mean conservation equation by a nondivergent, three-dimensional version of the quasi-Stokes velocity of T. J. McDougall and P. C. McIntosh, for which a closed analytical form for the streamfunction in terms of Eulerian mean quantities is given.

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Carsten Eden

Abstract

Following a suggestion by Tailleux, a consistent formulation of internal energy, the first law of thermodynamics, and the thermodynamic potentials for an ocean in Boussinesq approximation with a nonlinear equation of state is given. A modification of the pressure work in the first law is the only necessary modification from which all thermodynamic potentials and thermodynamic relations follow in a consistent way. This treatment of thermodynamics allows for a closed and explicit formulation of conservation equations for dynamic and potential reservoirs of both enthalpy and internal energy, which differentiate approximately reversible from irreversible effects on internal energy, and allows for a formulation of a closed energy cycle on which energetically consistent ocean models can be based on.

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Dirk Olbers and Carsten Eden

Abstract

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 Jürgen Willebrand

Abstract

A model of the Atlantic Ocean was forced with decadal-scale time series of surface fluxes taken from the National Centers for Environmental Prediction–National Center for Atmospheric Research reanalysis. The bulk of the variability of the oceanic circulation is found to be related to the North Atlantic oscillation (NAO). Both realistic experiments and idealized sensitivity studies with the model show a fast (intraseasonal timescale) barotropic response and a delayed (timescale about 6–8 yr) baroclinic oceanic response to the NAO. The fast response to a high NAO constitutes a barotropic anticyclonic circulation anomaly near the subpolar front with a substantial decrease of the northward heat transport and an increase of northward heat transport in the subtropics due to changes in Ekman transport. The delayed response is an increase in subpolar heat transport due to enhanced meridional overturning and due to a spinup of the subpolar gyre. The corresponding subpolar and subtropical heat content changes could in principle act as an immediate positive feedback and a delayed negative feedback to the NAO.

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Carsten Eden and Claus Böning

Abstract

Experiments with a suite of North Atlantic general circulation models are used to examine the sources of eddy kinetic energy (EKE) in the Labrador Sea. A high-resolution model version (1 12°) quantitatively reproduces the observed signature. A particular feature of the EKE in the Labrador Sea is its pronounced seasonal cycle, with a maximum intensity in early winter, as already found in earlier studies based on altimeter data. In contrast to a previously advanced hypothesis, the seasonally varying eddy field is not related to a forcing by high-frequency wind variations but can be explained by a seasonally modulated instability of the West Greenland Current (WGC). The main source of EKE in the Labrador Sea is an energy transfer due to Reynolds interaction work (barotropic instability) in a confined region near Cape Desolation where the WGC adjusts to a change in the topographic slope: Geostrophic contours tend to converge upstream of Cape Desolation, such that the topographically guided WGC narrows as well and becomes barotropically unstable. The eddies spawned from the WGC instability area, dominating the EKE in the interior Labrador Sea, are predominantly anticyclonic with warm and saline cores in the upper kilometer of the water column, while the few cyclones originating as well from the instability area show a more depth-independent structure. Companion experiments with a ⅓° model exhibit the strength of the WGC, influenced by either changes in the wind stress or heat flux forcing, as a leading factor determining seasonal to interannual changes of EKE in the Labrador Sea.

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Dirk Olbers and Carsten Eden

Abstract

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

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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Nils Brüggemann and Carsten Eden

Abstract

In this study, the authors discuss two different parameterizations for the effect of mixed layer eddies, one based on ageostrophic linear stability analysis (ALS) and the other one based on a scaling of the potential energy release by eddies (PER). Both parameterizations contradict each other in two aspects. First, they predict different functional relationships between the magnitude of the eddy fluxes and the Richardson number (Ri) related to the background state. Second, they also predict different vertical structure functions for the horizontal eddy fluxes. Numerical simulations for two different configurations and for a large range of different background conditions are used to evaluate the parameterizations. It turns out that PER is better suited to capture the Ri dependency of the magnitude of the eddy fluxes. On the other hand, the vertical structure of the meridional eddy fluxes predicted by ALS is more accurate than that of PER, while the vertical structure of the vertical eddy fluxes is well predicted by both parameterizations. Therefore, this study suggests the use of the magnitude of PER and the vertical structure functions of ALS for an improved parameterization of mixed layer eddy fluxes.

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Dirk Olbers and Carsten Eden

Abstract

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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