Energetics of the global ocean: The role of mesoscale eddies

This article reviews the energy cycle of the global ocean circulation, focusing on the role of baroclinic mesoscale eddies. Two of the important effects of mesoscale eddies are: (i) the flattening of the slope of large-scale isopycnal surfaces by the eddy-induced overturning circulation, the basis for the Gent–McWilliams parametrization; and (ii) the vertical redistribution of the momentum of basic geostrophic currents by the eddy-induced form stress (the residual effect of pressure perturbations), the basis for the Greatbatch–Lamb parametrization. While only point (i) can be explained using the classical Lorenz energy diagram, both (i) and (ii) can be explained using the modified energy diagram of Bleck as in the following energy cycle. Wind forcing provides an input to the mean KE, which is then transferred to the available potential energy (APE) of the large-scale field by the wind-induced Ekman flow. Subsequently, the APE is extracted by the eddy-induced overturning circulation to feed the mean KE, indicating the enhancement of the vertical shear of the basic current. Meanwhile, the vertical shear of the basic current is relaxed by the eddy-induced form stress, taking the mean KE to endow the eddy field with an energy cascade. The above energy cycle is useful for understanding the dynamics of the Antarctic Circumpolar Current. On the other hand, while the source of the eddy field energy has become clearer, identifying the sink and flux of the eddy field energy in both physical and spectral space remains major challenges of present-day oceanography. A recent study using a combination of models, satellite altimetry, and climatological hydrographic data shows that the western boundary acts as a “graveyard” for the westward-propagating eddies

The vertical structure of the surface wave radiation stress for circulation over a sloping bottom as given by thickness-weighted-mean theory

Previous attempts to derive the depth-dependent expression of the radiation stress have lead to a debate concerning (i) the applicability of Mellor’s approach to a sloping bottom, (ii) the introduction of the delta function at the mean sea surface in the later papers by Mellor, and (iii) a wave-induced pressure term derived in several recent studies. The authors use an equation system in vertically Lagrangian and horizontally Eulerian (VL) coordinates suitable for a concise treatment of the surface boundary, and obtain an expression for the depth-dependent radiation stress that is consistent with the vertically-integrated expression given by Longuet-Higgins and Stewart. Concerning (i)-(iii) in the above, the difficulty of handling a sloping bottom disappears when wave-averaged momentum equations in the VL coordinates are written for the development of (not the Lagrangian mean velocity but) the Eulerian mean velocity. There is also no delta function at the sea surface in the expression for the depth-dependent radiation stress. The connection between the wave-induced pressure term in the recent studies and the depth-dependent radiation stress term is easily shown by rewriting the pressure-based form stress term in the thickness-weighted-mean (TWM) momentum equations as a velocity-based term which contains the time derivative of the pseudomomentum in the TWM framework

The vertical structure of the surface wave radiation stress for circulation over a sloping bottom as given by thickness-weighted-mean theory

Previous attempts to derive the depth-dependent expression of the radiation stress have led to a debate concerning (i) the applicability of the Mellor approach to a sloping bottom, (ii) the introduction of the delta function at the mean sea surface in the later papers by Mellor, and (iii) a wave-induced pressure term derived in several recent studies. The authors use an equation system in vertically Lagrangian and horizontally Eulerian (VL) coordinates suitable for a concise treatment of the surface boundary and obtain an expression for the depth-dependent radiation stress that is consistent with the vertically integrated expression given by Longuet-Higgins and Stewart. Concerning (i)-(iii) above, the difficulty of handling a sloping bottom disappears when wave-averaged momentum equations in the VL coordinates are written for the development of (not the Lagrangian mean velocity but) the Eulerian mean velocity. There is also no delta function at the sea surface in the expression for the depth-dependent radiation stress. The connection between the wave-induced pressure term in the recent studies and the depth-dependent radiation stress term is easily shown by rewriting the pressure-based form stress term in the thickness-weighted-mean momentum equations as a velocity-based term that contains the time derivative of the pseudomomentum in the VL framework

An exact energy for TRM theory

ABSTRACT A time residual mean (TRM) energy is obtained by averaging a transformation of the energy of the Boussinesq hydrostatic incompressible equations of motion. The transformation is the fundamental TRM transformation between level Cartesian coordinates and coordinates that are the mean positions of density surfaces. The TRM energy consists of a sum of mean kinetic, mean potential, wave kinetic, and wave potential energies. It is shown that the interaction between the mean kinetic energy and mean potential energy can be expressed entirely in terms of mean fields. The wave forcing of the mean TRM momentum equations is expressed as a divergence. An explicit and exact form of the TRM equations, with the transformed pressure term expressed in terms of the mean and wave fields, is also noted. It is suggested that the mean domain for the TRM equations and the Cartesian domain may not be the same, which would have consequences for the TRM boundary conditions

A Divergence-Form Wave-Induced Pressure Inherent in the Extension of the Eliassen–Palm Theory to a Three-Dimensional Framework for All Waves at All Latitudes

Classical theory concerning theEliassen–Palmrelation is extended in this study to allowfor a unified treatment of midlatitude inertia–gravity waves (MIGWs), midlatitude Rossby waves (MRWs), and equatorial waves (EQWs). A conservation equation for what the authors call the impulse-bolus (IB) pseudomomentum is useful, because it is applicable to ageostrophic waves, and the associated three-dimensional flux is parallel to the direction of the group velocity of MRWs. The equation has previously been derived in an isentropic coordinate system or a shallow-water model. The authors make an explicit comparison of prognostic equations for the IB pseudomomentum vector and the classical energy-based (CE) pseudomomentum vector, assuming inviscid linear waves in a sufficiently weak mean flow, to provide a basis for the former quantity to be used in an Eulerian time-mean (EM) framework. The authors investigate what makes the three-dimensional fluxes in the IB and CE pseudomomentum equations look in different directions. It is found that the two fluxes are linked by a gauge transformation, previously unmentioned, associated with a divergence-form wave-induced pressure L. The quantity L vanishes for MIGWs and becomes nonzero for MRWs and EQWs, and it may be estimated using the virial theorem. Concerning the effect of waves on the mean flow, L represents an additional effect in the pressure gradient term of both (the three-dimensional versions of) the transformed EM momentum equations and the merged form of the EMmomentumequations, the latter of which is associated with the nonacceleration theorem

Editorial-The 4th International Workshop on Modeling the Ocean (IWMO 2012)

The 4th International Workshop on Modeling the Ocean (IWMO; http://www.jamstec.go.jp/frcgc/jcope/htdocs/e/ iwmo2012.html) was held on May 21–24, 2012 in the vibrant city of Yokohama on the Tokyo Bay, Japan. The Workshop was hosted by Japan Agency for Marine-Earth Science and Technology (JAMSTEC)—the home of the famous “Earth Simulator”—one of the world\u27s most powerful supercomputers dedicated for simulating the complex interactive processes of the earth and its environment

A new expression for the form stress term in the vertically Lagrangian mean framework for the effect of surface waves on the upper ocean circulation

There is an ongoing discussion in the community concerning the wave-averaged momentum equations in the hybrid vertically Lagrangian and horizontally Eulerian (VL) framework and, in particular, the form stress term (representing the residual effect of pressure perturbations) which is thought to restrict the handling of higher order waves in terms of a perturbation expansion. The present study shows that the traditional pressure-based form stress term can be transformed into a set of terms that do not contain any pressure quantities but do contain the time derivative of a wave-induced velocity. This wave-induced velocity is referred to as the pseudomomentum in the VL framework, as it is analogous to the generalized pseudomomentum in Andrews and McIntyre. This enables the second expression for the wave-averaged momentum equations in the VL framework (this time for the development of the total transport velocity minus the VL pseudomomentum) to be derived together with the vortex force. The velocity-based expression of the form stress term also contains the residual effect of the turbulent viscosity, which is useful for understanding the dissipation of wave energy leading to transfer of momentum from waves to circulation. It is found that the concept of the virtual wave stress of Longuet-Higgins is applicable to quite general situations: it does not matter whether there is wind forcing or not, the waves can have slow variations, and the viscosity coefficient can vary in the vertical. These results provide a basis for revisiting the surface boundary condition used in numerical circulation models

Thickness-weighted mean theory for the effect of surface gravity waves on mean flows in the upper ocean

The residual effect of surface gravity waves on mean flows in the upper ocean is investigated using thickness weighted mean (TWM) theory applied in a vertically Lagrangian and horizontally Eulerian coordinate system. Depth-dependent equations for the conservation of volume, momentum, and energy are derived. These equations allow for (i) finite amplitude fluid motions, (ii) the horizontal divergence of currents and (iii) a concise treatment of both the kinematic and viscous boundary conditions at the sea surface. Under the assumptions of steady and monochromatic waves and a uniform turbulent viscosity, the TWM momentum equations are used to illustrate the pressure- and viscosity-induced momentum fluxes through the surface that are implicit in previous studies of the wave-induced modification of the classical Ekman spiral problem. The TWM approach clarifies, in particular, the surface momentum flux associated with the so-called virtual wave stress of Longuet-Higgins. Overall the TWM framework can be regarded as an alternative to the three-dimensional Lagrangian mean framework of Pierson. Moreover the TWM framework can be used to include the residual effect of surface waves in large-scale circulation models. In specific models that carry the TWM velocity appropriate for advecting tracers as their velocity variable, the turbulent viscosity term should be modified so that the viscosity acts only on the Eulerian mean velocity